Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Move the limit into the exponent.

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

Step 1.1.2.1.3

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

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

Step 1.1.2.1.4

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

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

Step 1.1.2.1.5

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.1.2

Subtract $1$ from $1$ .

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

Step 1.1.2.3.1.3

Anything raised to $0$ is $1$ .

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

Step 1.1.2.3.1.4

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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Step 1.3.3.1

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 1.3.3.1.1

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

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

Step 1.3.3.1.2

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

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

Step 1.3.3.1.3

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Add $1$ and $0$ .

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

Step 1.3.3.6

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

Add $1$ and $0$ .

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

Step 1.4

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

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Multiply $- 1$ by $1$ .

$e^{1 - 1}$

Step 4.2

Subtract $1$ from $1$ .

$e^{0}$

Step 4.3

Anything raised to $0$ is $1$ .

$1$