Calculus Examples

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

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.

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

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

Step 1.2.2

Move the limit into the exponent.

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

Step 1.2.3

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

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.5

Simplify the answer.

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Step 1.2.5.1

Simplify each term.

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Step 1.2.5.1.1

Anything raised to $0$ is $1$ .

$\frac{1 + 0 - 1 \cdot 1}{lim_{x \to 0} \cos (x) - 1}$

Step 1.2.5.1.2

Multiply $- 1$ by $1$ .

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

Step 1.2.5.2

Add $1$ and $0$ .

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

Step 1.2.5.3

Subtract $1$ from $1$ .

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

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 $0$ .

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

Step 1.3.1.2

Move the limit inside the trig function because cosine is continuous.

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

Step 1.3.1.3

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

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

Step 1.3.2

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

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

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

The exact value of $\cos (0)$ 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $- 1$ by $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 3.9

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

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

Step 3.10

Add $- \sin (x)$ and $0$ .

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

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

Evaluate the limit.

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Step 4.1.2.1.1

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

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

Step 4.1.2.1.2

Move the limit into the exponent.

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

Step 4.1.2.1.3

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Simplify the answer.

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Step 4.1.2.3.1

Simplify each term.

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Step 4.1.2.3.1.1

Anything raised to $0$ is $1$ .

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

Step 4.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 4.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} - \sin (x)}$

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Evaluate the limit.

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Step 4.1.3.1.1

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

$\frac{0}{- lim_{x \to 0} \sin (x)}$

Step 4.1.3.1.2

Move the limit inside the trig function because sine is continuous.

$\frac{0}{- \sin (lim_{x \to 0} x)}$

Step 4.1.3.2

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

$\frac{0}{- \sin (0)}$

Step 4.1.3.3

Simplify the answer.

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Step 4.1.3.3.1

The exact value of $\sin (0)$ is $0$ .

$\frac{0}{- 0}$

Step 4.1.3.3.2

Multiply $- 1$ by $0$ .

$\frac{0}{0}$

Step 4.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 4.1.3.4

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

Undefined

$\frac{0}{0}$

Step 4.1.4

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

Undefined

$\frac{0}{0}$

Step 4.2

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

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

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

Step 4.3.7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

Move the limit into the exponent.

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

Step 5.3

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

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

Step 5.4

Move the limit inside the trig function because cosine is continuous.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

$\frac{e^{0}}{- \cos (0)}$

Step 7

Simplify the answer.

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Step 7.1

Separate fractions.

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

Step 7.2

Convert from $\frac{1}{\cos (0)}$ to $\sec (0)$ .

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

Step 7.3

Move the negative one from the denominator of $\frac{e^{0}}{- 1}$ .

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

Step 7.4

Rewrite $- 1 \cdot e^{0}$ as $- e^{0}$ .

$- e^{0} \sec (0)$

Step 7.5

Anything raised to $0$ is $1$ .

$- 1 \cdot 1 \sec (0)$

Step 7.6

Multiply $- 1$ by $1$ .

$- \sec (0)$

Step 7.7

The exact value of $\sec (0)$ is $1$ .

$- 1 \cdot 1$

Step 7.8

Multiply $- 1$ by $1$ .

$- 1$