Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(\cos (x) + \sin (x))}^{\frac{1}{x}}$ as $e^{\ln ({(\cos (x) + \sin (x))}^{\frac{1}{x}})}$ .

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

Step 1.2

Expand $\ln ({(\cos (x) + \sin (x))}^{\frac{1}{x}})$ by moving $\frac{1}{x}$ outside the logarithm.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

Combine $\frac{1}{x}$ and $\ln (\cos (x) + \sin (x))$ .

$e^{lim_{x \to 0} \frac{\ln (\cos (x) + \sin (x))}{x}}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$e^{\frac{lim_{x \to 0} \ln (\cos (x) + \sin (x))}{lim_{x \to 0} x}}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Move the limit inside the logarithm.

$e^{\frac{\ln (lim_{x \to 0} \cos (x) + \sin (x))}{lim_{x \to 0} x}}$

Step 3.1.2.2

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

$e^{\frac{\ln (lim_{x \to 0} \cos (x) + lim_{x \to 0} \sin (x))}{lim_{x \to 0} x}}$

Step 3.1.2.3

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

$e^{\frac{\ln (\cos (lim_{x \to 0} x) + lim_{x \to 0} \sin (x))}{lim_{x \to 0} x}}$

Step 3.1.2.4

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

$e^{\frac{\ln (\cos (lim_{x \to 0} x) + \sin (lim_{x \to 0} x))}{lim_{x \to 0} x}}$

Step 3.1.2.5

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

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Step 3.1.2.5.1

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

$e^{\frac{\ln (\cos (0) + \sin (lim_{x \to 0} x))}{lim_{x \to 0} x}}$

Step 3.1.2.5.2

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

$e^{\frac{\ln (\cos (0) + \sin (0))}{lim_{x \to 0} x}}$

Step 3.1.2.6

Simplify the answer.

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Step 3.1.2.6.1

Simplify each term.

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Step 3.1.2.6.1.1

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

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

Step 3.1.2.6.1.2

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

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

Step 3.1.2.6.2

Add $1$ and $0$ .

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

Step 3.1.2.6.3

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

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

Step 3.1.3

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

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

Step 3.1.4

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

Undefined

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

Step 3.2

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.

$lim_{x \to 0} \frac{\ln (\cos (x) + \sin (x))}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\cos (x) + \sin (x))]}{\frac{d}{d x} [x]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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 $g (x) = \cos (x) + \sin (x)$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $\cos (x) + \sin (x)$ .

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Replace all occurrences of $u$ with $\cos (x) + \sin (x)$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Simplify.

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Step 3.3.6.1

Reorder the factors of $\frac{1}{\cos (x) + \sin (x)} (- \sin (x) + \cos (x))$ .

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

Step 3.3.6.2

Apply the distributive property.

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

Step 3.3.6.3

Combine $\frac{1}{\cos (x) + \sin (x)}$ and $\sin (x)$ .

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

Step 3.3.6.4

Combine $\cos (x)$ and $\frac{1}{\cos (x) + \sin (x)}$ .

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

Step 3.3.6.5

Combine the numerators over the common denominator.

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

Step 3.3.6.6

Factor $- 1$ out of $- \sin (x)$ .

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

Step 3.3.6.7

Factor $- 1$ out of $\cos (x)$ .

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

Step 3.3.6.8

Factor $- 1$ out of $- (\sin (x)) - 1 (- \cos (x))$ .

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

Step 3.3.6.9

Rewrite $- (\sin (x) - \cos (x))$ as $- 1 (\sin (x) - \cos (x))$ .

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

Step 3.3.6.10

Move the negative in front of the fraction.

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

Step 3.3.7

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Multiply $- 1$ by $1$ .

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Multiply $- 1$ by $1$ .

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

Step 6.1.4

Subtract $1$ from $0$ .

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

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Add $1$ and $0$ .

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

Step 6.3

Divide $- 1$ by $1$ .

$e^{- - 1}$

Step 6.4

Multiply $- 1$ by $- 1$ .

$e^{1}$

Step 6.5

Simplify.

$e$

Step 7

The result can be shown in multiple forms.

Exact Form:

$e$

Decimal Form:

$2.71828182 \dots$