Calculus Examples

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

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.

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

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 $0$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

Move the limit into the exponent.

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

Step 1.1.2.2

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

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

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

Anything raised to $0$ is $1$ .

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

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

Move the limit inside the logarithm.

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

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

Step 1.1.3.1.4

Move the limit into the exponent.

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

Step 1.1.3.2

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

$\frac{0}{\ln (2 - e^{0})}$

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

Anything raised to $0$ is $1$ .

$\frac{0}{\ln (2 - 1 \cdot 1)}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $1$ .

$\frac{0}{\ln (2 - 1)}$

Step 1.1.3.3.2

Subtract $1$ from $2$ .

$\frac{0}{\ln (1)}$

Step 1.1.3.3.3

The natural logarithm of $1$ is $0$ .

$\frac{0}{0}$

Step 1.1.3.3.4

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.

$lim_{x \to 0} \frac{1 - e^{x}}{\ln (2 - e^{x})} = lim_{x \to 0} \frac{\frac{d}{d x} [1 - e^{x}]}{\frac{d}{d x} [\ln (2 - e^{x})]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 0} \frac{0 - \frac{d}{d x} [e^{x}]}{\frac{d}{d x} [\ln (2 - e^{x})]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{0 - e^{x}}{\frac{d}{d x} [\ln (2 - e^{x})]}$

Step 1.3.5

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

$lim_{x \to 0} \frac{- e^{x}}{\frac{d}{d x} [\ln (2 - e^{x})]}$

Step 1.3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 2 - e^{x}$ .

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Step 1.3.6.1

To apply the Chain Rule, set $u$ as $2 - e^{x}$ .

$lim_{x \to 0} \frac{- e^{x}}{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [2 - e^{x}]}$

Step 1.3.6.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 1.3.6.3

Replace all occurrences of $u$ with $2 - e^{x}$ .

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

Step 1.3.7

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

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

Step 1.3.8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 1.3.9

Add $0$ and $\frac{d}{d x} [- e^{x}]$ .

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

Step 1.3.10

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

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

Step 1.3.11

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

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

Step 1.3.12

Combine $\frac{1}{2 - e^{x}}$ and $e^{x}$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Combine factors.

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Step 1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.5.2

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

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

Step 1.5.3

Combine $e^{x}$ and $\frac{2 - e^{x}}{e^{x}}$ .

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

Step 1.6

Cancel the common factor of $e^{x}$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Divide $2 - e^{x}$ by $1$ .

$lim_{x \to 0} 2 - e^{x}$

Step 2

Evaluate the limit.

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Step 2.1

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

$lim_{x \to 0} 2 - lim_{x \to 0} e^{x}$

Step 2.2

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$2 - lim_{x \to 0} e^{x}$

Step 2.3

Move the limit into the exponent.

$2 - e^{lim_{x \to 0} x}$

Step 3

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

$2 - e^{0}$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Anything raised to $0$ is $1$ .

$2 - 1 \cdot 1$

Step 4.1.2

Multiply $- 1$ by $1$ .

$2 - 1$

Step 4.2

Subtract $1$ from $2$ .

$1$