Calculus Examples

$lim_{x \to 1} \frac{x e^{x} - e}{x^{2} - 1}$

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.

$\frac{lim_{x \to 1} x e^{x} - e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 1} x e^{x} - lim_{x \to 1} e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.2

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

$\frac{lim_{x \to 1} x \cdot lim_{x \to 1} e^{x} - lim_{x \to 1} e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.3

Move the limit into the exponent.

$\frac{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x} - lim_{x \to 1} e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.4

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

$\frac{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x} - e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

$\frac{1 \cdot e^{lim_{x \to 1} x} - e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.5.2

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

$\frac{1 \cdot e^{1} - e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

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

$\frac{e^{1} - e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.6.1.2

Simplify.

$\frac{e - e}{lim_{x \to 1} x^{2} - 1}$

Step 1.1.2.6.2

Subtract $e$ from $e$ .

$\frac{0}{lim_{x \to 1} x^{2} - 1}$

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 $1$ .

$\frac{0}{lim_{x \to 1} x^{2} - lim_{x \to 1} 1}$

Step 1.1.3.1.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1}$

Step 1.1.3.1.3

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

$\frac{0}{{(lim_{x \to 1} x)}^{2} - 1 \cdot 1}$

Step 1.1.3.2

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

$\frac{0}{1^{2} - 1 \cdot 1}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

One to any power is one.

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

Step 1.1.3.3.1.2

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.

$lim_{x \to 1} \frac{x e^{x} - e}{x^{2} - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [x e^{x} - e]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [x e^{x} - e]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.2

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

$lim_{x \to 1} \frac{\frac{d}{d x} [x e^{x}] + \frac{d}{d x} [- e]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.3

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

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Step 1.3.3.1

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 $g (x) = e^{x}$ .

$lim_{x \to 1} \frac{x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x] + \frac{d}{d x} [- e]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.3.2

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

$lim_{x \to 1} \frac{x e^{x} + e^{x} \frac{d}{d x} [x] + \frac{d}{d x} [- e]}{\frac{d}{d x} [x^{2} - 1]}$

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 $n = 1$ .

$lim_{x \to 1} \frac{x e^{x} + e^{x} \cdot 1 + \frac{d}{d x} [- e]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.3.4

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

$lim_{x \to 1} \frac{x e^{x} + e^{x} + \frac{d}{d x} [- e]}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.4

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

$lim_{x \to 1} \frac{x e^{x} + e^{x} + 0}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

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

$lim_{x \to 1} \frac{x e^{x} + e^{x}}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.5.2

Reorder terms.

$lim_{x \to 1} \frac{e^{x} x + e^{x}}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.5.3

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

$lim_{x \to 1} \frac{x e^{x} + e^{x}}{\frac{d}{d x} [x^{2} - 1]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{x e^{x} + e^{x}}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]}$

Step 1.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 1} \frac{x e^{x} + e^{x}}{2 x + \frac{d}{d x} [- 1]}$

Step 1.3.8

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

$lim_{x \to 1} \frac{x e^{x} + e^{x}}{2 x + 0}$

Step 1.3.9

Add $2 x$ and $0$ .

$lim_{x \to 1} \frac{x e^{x} + e^{x}}{2 x}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 1} \frac{x e^{x} + e^{x}}{x}$

Step 2.2

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

$\frac{1}{2} \cdot \frac{lim_{x \to 1} x e^{x} + e^{x}}{lim_{x \to 1} x}$

Step 2.3

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

$\frac{1}{2} \cdot \frac{lim_{x \to 1} x e^{x} + lim_{x \to 1} e^{x}}{lim_{x \to 1} x}$

Step 2.4

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

$\frac{1}{2} \cdot \frac{lim_{x \to 1} x \cdot lim_{x \to 1} e^{x} + lim_{x \to 1} e^{x}}{lim_{x \to 1} x}$

Step 2.5

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x} + lim_{x \to 1} e^{x}}{lim_{x \to 1} x}$

Step 2.6

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{lim_{x \to 1} x \cdot e^{lim_{x \to 1} x} + e^{lim_{x \to 1} x}}{lim_{x \to 1} x}$

Step 3

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

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Step 3.1

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

$\frac{1}{2} \cdot \frac{1 \cdot e^{lim_{x \to 1} x} + e^{lim_{x \to 1} x}}{lim_{x \to 1} x}$

Step 3.2

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

$\frac{1}{2} \cdot \frac{1 \cdot e^{1} + e^{lim_{x \to 1} x}}{lim_{x \to 1} x}$

Step 3.3

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

$\frac{1}{2} \cdot \frac{1 \cdot e^{1} + e^{1}}{lim_{x \to 1} x}$

Step 3.4

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

$\frac{1}{2} \cdot \frac{1 \cdot e^{1} + e^{1}}{1}$

Step 4

Simplify the answer.

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Step 4.1

Divide $1 \cdot e^{1} + e^{1}$ by $1$ .

$\frac{1}{2} (1 \cdot e^{1} + e^{1})$

Step 4.2

Simplify each term.

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Step 4.2.1

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

$\frac{1}{2} (e^{1} + e^{1})$

Step 4.2.2

Simplify.

$\frac{1}{2} (e + e^{1})$

Step 4.2.3

Simplify.

$\frac{1}{2} (e + e)$

Step 4.3

Add $e$ and $e$ .

$\frac{1}{2} (2 e)$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Factor $2$ out of $2 e$ .

$\frac{1}{2} (2 (e))$

Step 4.4.2

Cancel the common factor.

$\frac{1}{2} (2 e)$

Step 4.4.3

Rewrite the expression.

$e$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e$

Decimal Form:

$2.71828182 \dots$