Calculus Examples

$lim_{x \to 1} \frac{1}{e^{x - 1} - 1} - \frac{1}{x - 1}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{e^{x - 1} - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$lim_{x \to 1} \frac{1}{e^{x - 1} - 1} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1}$

Step 1.2

To write $- \frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{e^{x - 1} - 1}{e^{x - 1} - 1}$ .

$lim_{x \to 1} \frac{1}{e^{x - 1} - 1} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1} \cdot \frac{e^{x - 1} - 1}{e^{x - 1} - 1}$

Step 1.3

Write each expression with a common denominator of $(e^{x - 1} - 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{e^{x - 1} - 1}$ by $\frac{x - 1}{x - 1}$ .

$lim_{x \to 1} \frac{x - 1}{(e^{x - 1} - 1) (x - 1)} - \frac{1}{x - 1} \cdot \frac{e^{x - 1} - 1}{e^{x - 1} - 1}$

Step 1.3.2

Multiply $\frac{1}{x - 1}$ by $\frac{e^{x - 1} - 1}{e^{x - 1} - 1}$ .

$lim_{x \to 1} \frac{x - 1}{(e^{x - 1} - 1) (x - 1)} - \frac{e^{x - 1} - 1}{(x - 1) (e^{x - 1} - 1)}$

Step 1.3.3

Reorder the factors of $(x - 1) (e^{x - 1} - 1)$ .

$lim_{x \to 1} \frac{x - 1}{(e^{x - 1} - 1) (x - 1)} - \frac{e^{x - 1} - 1}{(e^{x - 1} - 1) (x - 1)}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{x - 1 - (e^{x - 1} - 1)}{(e^{x - 1} - 1) (x - 1)}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 1} x - 1 - (e^{x - 1} - 1)}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.2

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 2.1.2.1

Evaluate the limit of $x - 1 - (e^{x - 1} - 1)$ by plugging in $1$ for $x$ .

$\frac{1 - 1 - (e^{1 - 1} - 1)}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.2.2

Simplify each term.

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Step 2.1.2.2.1

Simplify each term.

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Step 2.1.2.2.1.1

Subtract $1$ from $1$ .

$\frac{1 - 1 - (e^{0} - 1)}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.2.2.1.2

Anything raised to $0$ is $1$ .

$\frac{1 - 1 - (1 - 1)}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.2.2.2

Subtract $1$ from $1$ .

$\frac{1 - 1 - 0}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.2.2.3

Multiply $- 1$ by $0$ .

$\frac{1 - 1 + 0}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.2.3

Subtract $1$ from $1$ .

$\frac{0 + 0}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.2.4

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 1} (e^{x - 1} - 1) (x - 1)}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} e^{x - 1} - 1 \cdot lim_{x \to 1} x - 1}$

Step 2.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{(lim_{x \to 1} e^{x - 1} - lim_{x \to 1} 1) \cdot lim_{x \to 1} x - 1}$

Step 2.1.3.3

Move the limit into the exponent.

$\frac{0}{(e^{lim_{x \to 1} x - 1} - lim_{x \to 1} 1) \cdot lim_{x \to 1} x - 1}$

Step 2.1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{(e^{lim_{x \to 1} x - lim_{x \to 1} 1} - lim_{x \to 1} 1) \cdot lim_{x \to 1} x - 1}$

Step 2.1.3.5

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{0}{(e^{lim_{x \to 1} x - 1 \cdot 1} - lim_{x \to 1} 1) \cdot lim_{x \to 1} x - 1}$

Step 2.1.3.6

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{0}{(e^{lim_{x \to 1} x - 1 \cdot 1} - 1 \cdot 1) \cdot lim_{x \to 1} x - 1}$

Step 2.1.3.7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{(e^{lim_{x \to 1} x - 1 \cdot 1} - 1 \cdot 1) \cdot (lim_{x \to 1} x - lim_{x \to 1} 1)}$

Step 2.1.3.8

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{0}{(e^{lim_{x \to 1} x - 1 \cdot 1} - 1 \cdot 1) \cdot (lim_{x \to 1} x - 1 \cdot 1)}$

Step 2.1.3.9

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 2.1.3.9.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{(e^{1 - 1 \cdot 1} - 1 \cdot 1) \cdot (lim_{x \to 1} x - 1 \cdot 1)}$

Step 2.1.3.9.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{(e^{1 - 1 \cdot 1} - 1 \cdot 1) \cdot (1 - 1 \cdot 1)}$

Step 2.1.3.10

Simplify the answer.

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Step 2.1.3.10.1

Simplify each term.

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Step 2.1.3.10.1.1

Multiply $- 1$ by $1$ .

$\frac{0}{(e^{1 - 1} - 1 \cdot 1) \cdot (1 - 1 \cdot 1)}$

Step 2.1.3.10.1.2

Subtract $1$ from $1$ .

$\frac{0}{(e^{0} - 1 \cdot 1) \cdot (1 - 1 \cdot 1)}$

Step 2.1.3.10.1.3

Anything raised to $0$ is $1$ .

$\frac{0}{(1 - 1 \cdot 1) \cdot (1 - 1 \cdot 1)}$

Step 2.1.3.10.1.4

Multiply $- 1$ by $1$ .

$\frac{0}{(1 - 1) \cdot (1 - 1 \cdot 1)}$

Step 2.1.3.10.2

Subtract $1$ from $1$ .

$\frac{0}{0 \cdot (1 - 1 \cdot 1)}$

Step 2.1.3.10.3

Multiply $- 1$ by $1$ .

$\frac{0}{0 \cdot (1 - 1)}$

Step 2.1.3.10.4

Subtract $1$ from $1$ .

$\frac{0}{0 \cdot 0}$

Step 2.1.3.10.5

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 2.1.3.10.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.3.11

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 1} \frac{x - 1 - (e^{x - 1} - 1)}{(e^{x - 1} - 1) (x - 1)} = lim_{x \to 1} \frac{\frac{d}{d x} [x - 1 - (e^{x - 1} - 1)]}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [x - 1 - (e^{x - 1} - 1)]}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.2

By the Sum Rule, the derivative of $x - 1 - (e^{x - 1} - 1)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (e^{x - 1} - 1)]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (e^{x - 1} - 1)]}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{1 + \frac{d}{d x} [- 1] + \frac{d}{d x} [- (e^{x - 1} - 1)]}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{1 + 0 + \frac{d}{d x} [- (e^{x - 1} - 1)]}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5

Evaluate $\frac{d}{d x} [- (e^{x - 1} - 1)]$ .

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Step 2.3.5.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (e^{x - 1} - 1)$ with respect to $x$ is $- \frac{d}{d x} [e^{x - 1} - 1]$ .

$lim_{x \to 1} \frac{1 + 0 - \frac{d}{d x} [e^{x - 1} - 1]}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.2

By the Sum Rule, the derivative of $e^{x - 1} - 1$ with respect to $x$ is $\frac{d}{d x} [e^{x - 1}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{1 + 0 - (\frac{d}{d x} [e^{x - 1}] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x - 1$ .

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Step 2.3.5.3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$lim_{x \to 1} \frac{1 + 0 - (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x - 1] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{u} \frac{d}{d x} [x - 1] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.3.3

Replace all occurrences of $u$ with $x - 1$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{x - 1} \frac{d}{d x} [x - 1] + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.4

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{x - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{x - 1} (1 + \frac{d}{d x} [- 1]) + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{x - 1} (1 + 0) + \frac{d}{d x} [- 1])}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{x - 1} (1 + 0) + 0)}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.8

Add $1$ and $0$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{x - 1} \cdot 1 + 0)}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.9

Multiply $e^{x - 1}$ by $1$ .

$lim_{x \to 1} \frac{1 + 0 - (e^{x - 1} + 0)}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.5.10

Add $e^{x - 1}$ and $0$ .

$lim_{x \to 1} \frac{1 + 0 - e^{x - 1}}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.6

Add $1$ and $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{\frac{d}{d x} [(e^{x - 1} - 1) (x - 1)]}$

Step 2.3.7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x - 1} - 1$ and $g (x) = x - 1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{(e^{x - 1} - 1) \frac{d}{d x} [x - 1] + (x - 1) \frac{d}{d x} [e^{x - 1} - 1]}$

Step 2.3.8

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{(e^{x - 1} - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [e^{x - 1} - 1]}$

Step 2.3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{(e^{x - 1} - 1) (1 + \frac{d}{d x} [- 1]) + (x - 1) \frac{d}{d x} [e^{x - 1} - 1]}$

Step 2.3.10

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{(e^{x - 1} - 1) (1 + 0) + (x - 1) \frac{d}{d x} [e^{x - 1} - 1]}$

Step 2.3.11

Add $1$ and $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{(e^{x - 1} - 1) \cdot 1 + (x - 1) \frac{d}{d x} [e^{x - 1} - 1]}$

Step 2.3.12

Multiply $e^{x - 1} - 1$ by $1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) \frac{d}{d x} [e^{x - 1} - 1]}$

Step 2.3.13

By the Sum Rule, the derivative of $e^{x - 1} - 1$ with respect to $x$ is $\frac{d}{d x} [e^{x - 1}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (\frac{d}{d x} [e^{x - 1}] + \frac{d}{d x} [- 1])}$

Step 2.3.14

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x - 1$ .

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Step 2.3.14.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x - 1] + \frac{d}{d x} [- 1])}$

Step 2.3.14.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{u} \frac{d}{d x} [x - 1] + \frac{d}{d x} [- 1])}$

Step 2.3.14.3

Replace all occurrences of $u$ with $x - 1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{x - 1} \frac{d}{d x} [x - 1] + \frac{d}{d x} [- 1])}$

Step 2.3.15

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{x - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + \frac{d}{d x} [- 1])}$

Step 2.3.16

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{x - 1} (1 + \frac{d}{d x} [- 1]) + \frac{d}{d x} [- 1])}$

Step 2.3.17

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{x - 1} (1 + 0) + \frac{d}{d x} [- 1])}$

Step 2.3.18

Add $1$ and $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{x - 1} \cdot 1 + \frac{d}{d x} [- 1])}$

Step 2.3.19

Multiply $e^{x - 1}$ by $1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{x - 1} + \frac{d}{d x} [- 1])}$

Step 2.3.20

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) (e^{x - 1} + 0)}$

Step 2.3.21

Add $e^{x - 1}$ and $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} - 1 + (x - 1) e^{x - 1}}$

Step 2.3.22

Simplify.

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Step 2.3.22.1

Reorder terms.

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} (x - 1) + e^{x - 1} - 1}$

Step 2.3.22.2

Simplify each term.

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Step 2.3.22.2.1

Apply the distributive property.

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} x + e^{x - 1} \cdot - 1 + e^{x - 1} - 1}$

Step 2.3.22.2.2

Move $- 1$ to the left of $e^{x - 1}$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} x - 1 \cdot e^{x - 1} + e^{x - 1} - 1}$

Step 2.3.22.2.3

Rewrite $- 1 e^{x - 1}$ as $- e^{x - 1}$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} x - e^{x - 1} + e^{x - 1} - 1}$

Step 2.3.22.3

Combine the opposite terms in $e^{x - 1} x - e^{x - 1} + e^{x - 1} - 1$ .

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Step 2.3.22.3.1

Add $- e^{x - 1}$ and $e^{x - 1}$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} x + 0 - 1}$

Step 2.3.22.3.2

Add $e^{x - 1} x$ and $0$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{e^{x - 1} x - 1}$

Step 2.3.22.4

Reorder factors in $e^{x - 1} x - 1$ .

$lim_{x \to 1} \frac{1 - e^{x - 1}}{x e^{x - 1} - 1}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 1} 1 - e^{x - 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 1 - lim_{x \to 1} e^{x - 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{1 - lim_{x \to 1} e^{x - 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.1.3

Move the limit into the exponent.

$\frac{1 - e^{lim_{x \to 1} x - 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.1.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{1 - e^{lim_{x \to 1} x - lim_{x \to 1} 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{1 - e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1 - e^{1 - 1 \cdot 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Multiply $- 1$ by $1$ .

$\frac{1 - e^{1 - 1}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.3.1.2

Subtract $1$ from $1$ .

$\frac{1 - e^{0}}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.3.1.3

Anything raised to $0$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.3.1.4

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} x e^{x - 1} - 1}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x e^{x - 1} - lim_{x \to 1} 1}$

Step 3.1.3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x \cdot lim_{x \to 1} e^{x - 1} - lim_{x \to 1} 1}$

Step 3.1.3.3

Move the limit into the exponent.

$\frac{0}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1} - lim_{x \to 1} 1}$

Step 3.1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - lim_{x \to 1} 1} - lim_{x \to 1} 1}$

Step 3.1.3.5

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1 \cdot 1} - lim_{x \to 1} 1}$

Step 3.1.3.6

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1 \cdot 1} - 1 \cdot 1}$

Step 3.1.3.7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 3.1.3.7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{1 \cdot e^{lim_{x \to 1} x - 1 \cdot 1} - 1 \cdot 1}$

Step 3.1.3.7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{1 \cdot e^{1 - 1 \cdot 1} - 1 \cdot 1}$

Step 3.1.3.8

Simplify the answer.

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Step 3.1.3.8.1

Simplify each term.

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Step 3.1.3.8.1.1

Multiply $e^{1 - 1 \cdot 1}$ by $1$ .

$\frac{0}{e^{1 - 1 \cdot 1} - 1 \cdot 1}$

Step 3.1.3.8.1.2

Multiply $- 1$ by $1$ .

$\frac{0}{e^{1 - 1} - 1 \cdot 1}$

Step 3.1.3.8.1.3

Subtract $1$ from $1$ .

$\frac{0}{e^{0} - 1 \cdot 1}$

Step 3.1.3.8.1.4

Anything raised to $0$ is $1$ .

$\frac{0}{1 - 1 \cdot 1}$

Step 3.1.3.8.1.5

Multiply $- 1$ by $1$ .

$\frac{0}{1 - 1}$

Step 3.1.3.8.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 3.1.3.8.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.3.9

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 1} \frac{1 - e^{x - 1}}{x e^{x - 1} - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [1 - e^{x - 1}]}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [1 - e^{x - 1}]}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.2

By the Sum Rule, the derivative of $1 - e^{x - 1}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x - 1}]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x - 1}]}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{0 + \frac{d}{d x} [- e^{x - 1}]}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [- e^{x - 1}]$ .

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Step 3.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{x - 1}$ with respect to $x$ is $- \frac{d}{d x} [e^{x - 1}]$ .

$lim_{x \to 1} \frac{0 - \frac{d}{d x} [e^{x - 1}]}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x - 1$ .

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Step 3.3.4.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$lim_{x \to 1} \frac{0 - (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x - 1])}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 1} \frac{0 - (e^{u} \frac{d}{d x} [x - 1])}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.2.3

Replace all occurrences of $u$ with $x - 1$ .

$lim_{x \to 1} \frac{0 - (e^{x - 1} \frac{d}{d x} [x - 1])}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.3

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{0 - (e^{x - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]))}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{0 - (e^{x - 1} (1 + \frac{d}{d x} [- 1]))}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{0 - (e^{x - 1} (1 + 0))}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.6

Add $1$ and $0$ .

$lim_{x \to 1} \frac{0 - (e^{x - 1} \cdot 1)}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.4.7

Multiply $e^{x - 1}$ by $1$ .

$lim_{x \to 1} \frac{0 - e^{x - 1}}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.5

Subtract $e^{x - 1}$ from $0$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{\frac{d}{d x} [x e^{x - 1} - 1]}$

Step 3.3.6

By the Sum Rule, the derivative of $x e^{x - 1} - 1$ with respect to $x$ is $\frac{d}{d x} [x e^{x - 1}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{\frac{d}{d x} [x e^{x - 1}] + \frac{d}{d x} [- 1]}$

Step 3.3.7

Evaluate $\frac{d}{d x} [x e^{x - 1}]$ .

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Step 3.3.7.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x - 1}$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x \frac{d}{d x} [e^{x - 1}] + e^{x - 1} \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 3.3.7.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x - 1$ .

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Step 3.3.7.2.1

To apply the Chain Rule, set $u$ as $x - 1$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x - 1]) + e^{x - 1} \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 3.3.7.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (e^{u} \frac{d}{d x} [x - 1]) + e^{x - 1} \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 3.3.7.2.3

Replace all occurrences of $u$ with $x - 1$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (e^{x - 1} \frac{d}{d x} [x - 1]) + e^{x - 1} \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 3.3.7.3

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (e^{x - 1} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])) + e^{x - 1} \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 3.3.7.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (e^{x - 1} (1 + \frac{d}{d x} [- 1])) + e^{x - 1} \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 3.3.7.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (e^{x - 1} (1 + 0)) + e^{x - 1} \frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 3.3.7.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (e^{x - 1} (1 + 0)) + e^{x - 1} \cdot 1 + \frac{d}{d x} [- 1]}$

Step 3.3.7.7

Add $1$ and $0$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x (e^{x - 1} \cdot 1) + e^{x - 1} \cdot 1 + \frac{d}{d x} [- 1]}$

Step 3.3.7.8

Multiply $e^{x - 1}$ by $1$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x e^{x - 1} + e^{x - 1} \cdot 1 + \frac{d}{d x} [- 1]}$

Step 3.3.7.9

Multiply $e^{x - 1}$ by $1$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x e^{x - 1} + e^{x - 1} + \frac{d}{d x} [- 1]}$

Step 3.3.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x e^{x - 1} + e^{x - 1} + 0}$

Step 3.3.9

Simplify.

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Step 3.3.9.1

Add $x e^{x - 1} + e^{x - 1}$ and $0$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x e^{x - 1} + e^{x - 1}}$

Step 3.3.9.2

Reorder terms.

$lim_{x \to 1} \frac{- e^{x - 1}}{e^{x - 1} x + e^{x - 1}}$

Step 3.3.9.3

Reorder factors in $e^{x - 1} x + e^{x - 1}$ .

$lim_{x \to 1} \frac{- e^{x - 1}}{x e^{x - 1} + e^{x - 1}}$

Step 4

Evaluate the limit.

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Step 4.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} - e^{x - 1}}{lim_{x \to 1} x e^{x - 1} + e^{x - 1}}$

Step 4.2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{- lim_{x \to 1} e^{x - 1}}{lim_{x \to 1} x e^{x - 1} + e^{x - 1}}$

Step 4.3

Move the limit into the exponent.

$\frac{- e^{lim_{x \to 1} x - 1}}{lim_{x \to 1} x e^{x - 1} + e^{x - 1}}$

Step 4.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - lim_{x \to 1} 1}}{lim_{x \to 1} x e^{x - 1} + e^{x - 1}}$

Step 4.5

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x e^{x - 1} + e^{x - 1}}$

Step 4.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x e^{x - 1} + lim_{x \to 1} e^{x - 1}}$

Step 4.7

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x \cdot lim_{x \to 1} e^{x - 1} + lim_{x \to 1} e^{x - 1}}$

Step 4.8

Move the limit into the exponent.

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1} + lim_{x \to 1} e^{x - 1}}$

Step 4.9

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - lim_{x \to 1} 1} + lim_{x \to 1} e^{x - 1}}$

Step 4.10

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1 \cdot 1} + lim_{x \to 1} e^{x - 1}}$

Step 4.11

Move the limit into the exponent.

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1 \cdot 1} + e^{lim_{x \to 1} x - 1}}$

Step 4.12

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1 \cdot 1} + e^{lim_{x \to 1} x - lim_{x \to 1} 1}}$

Step 4.13

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{- e^{lim_{x \to 1} x - 1 \cdot 1}}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1 \cdot 1} + e^{lim_{x \to 1} x - 1 \cdot 1}}$

Step 5

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{- e^{1 - 1 \cdot 1}}{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x - 1 \cdot 1} + e^{lim_{x \to 1} x - 1 \cdot 1}}$

Step 5.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{- e^{1 - 1 \cdot 1}}{1 \cdot e^{lim_{x \to 1} x - 1 \cdot 1} + e^{lim_{x \to 1} x - 1 \cdot 1}}$

Step 5.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{- e^{1 - 1 \cdot 1}}{1 \cdot e^{1 - 1 \cdot 1} + e^{lim_{x \to 1} x - 1 \cdot 1}}$

Step 5.4

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{- e^{1 - 1 \cdot 1}}{1 \cdot e^{1 - 1 \cdot 1} + e^{1 - 1 \cdot 1}}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 1$ by $1$ .

$\frac{- e^{1 - 1}}{1 \cdot e^{1 - 1 \cdot 1} + e^{1 - 1 \cdot 1}}$

Step 6.1.2

Subtract $1$ from $1$ .

$\frac{- e^{0}}{1 \cdot e^{1 - 1 \cdot 1} + e^{1 - 1 \cdot 1}}$

Step 6.1.3

Anything raised to $0$ is $1$ .

$\frac{- 1 \cdot 1}{1 \cdot e^{1 - 1 \cdot 1} + e^{1 - 1 \cdot 1}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $e^{1 - 1 \cdot 1}$ by $1$ .

$\frac{- 1 \cdot 1}{e^{1 - 1 \cdot 1} + e^{1 - 1 \cdot 1}}$

Step 6.2.2

Multiply $- 1$ by $1$ .

$\frac{- 1 \cdot 1}{e^{1 - 1} + e^{1 - 1 \cdot 1}}$

Step 6.2.3

Subtract $1$ from $1$ .

$\frac{- 1 \cdot 1}{e^{0} + e^{1 - 1 \cdot 1}}$

Step 6.2.4

Anything raised to $0$ is $1$ .

$\frac{- 1 \cdot 1}{1 + e^{1 - 1 \cdot 1}}$

Step 6.2.5

Multiply $- 1$ by $1$ .

$\frac{- 1 \cdot 1}{1 + e^{1 - 1}}$

Step 6.2.6

Subtract $1$ from $1$ .

$\frac{- 1 \cdot 1}{1 + e^{0}}$

Step 6.2.7

Anything raised to $0$ is $1$ .

$\frac{- 1 \cdot 1}{1 + 1}$

Step 6.2.8

Add $1$ and $1$ .

$\frac{- 1 \cdot 1}{2}$

Step 6.3

Multiply $- 1$ by $1$ .

$\frac{- 1}{2}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$