Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 1.2.2

Move the limit into the exponent.

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

Step 1.2.3

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.2.3.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 1.2.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 1.2.4

Simplify the answer.

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Step 1.2.4.1

Anything raised to $0$ is $1$ .

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

Step 1.2.4.2

Multiply $0$ by $1$ .

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 1.3.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 1.3.1.3

Move the limit into the exponent.

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

Step 1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Anything raised to $0$ is $1$ .

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

Step 1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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Step 3.8.1

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

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

Step 3.8.2

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

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

Step 3.9

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

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

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

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

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

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

Step 7

Move the limit into the exponent.

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

Step 8

Move the limit into the exponent.

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

Step 9

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

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

Step 10

Move the limit into the exponent.

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

Step 11

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 11.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 11.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 11.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

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

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Anything raised to $0$ is $1$ .

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

Step 12.1.2

Multiply $0$ by $1$ .

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

Step 12.1.3

Anything raised to $0$ is $1$ .

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

Step 12.1.4

Add $0$ and $1$ .

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

Step 12.2

Anything raised to $0$ is $1$ .

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

Step 12.3

Multiply $- 1$ by $1$ .

$\frac{1}{- 1}$

Step 12.4

Divide $1$ by $- 1$ .

$- 1$