Calculus Examples

$lim_{x \to 0} {(1 + x)}^{\cot (x)}$

Step 1

Change the two-sided limit into a right sided limit.

$lim_{x \to 0^{+}} {(1 + x)}^{\cot (x)}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(1 + x)}^{\cot (x)}$ as $e^{\ln ({(1 + x)}^{\cot (x)})}$ .

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

Step 2.2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

Rewrite $\cot (x) \ln (1 + x)$ as $\frac{\ln (1 + x)}{\frac{1}{\cot (x)}}$ .

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

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

Evaluate the limit.

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Step 5.1.2.1.1

Move the limit inside the logarithm.

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

Step 5.1.2.1.2

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

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

Step 5.1.2.1.3

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

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

Step 5.1.2.2

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

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

Step 5.1.2.3

Simplify the answer.

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Step 5.1.2.3.1

Add $1$ and $0$ .

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

Step 5.1.2.3.2

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

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

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

Apply trigonometric identities.

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Step 5.1.3.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 5.1.3.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 5.1.3.1.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

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

$e^{\frac{0}{\tan (0)}}$

Step 5.1.3.4

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

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

Step 5.1.3.5

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

Undefined

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

Step 5.1.4

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

Undefined

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

Step 5.2

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.

$lim_{x \to 0^{+}} \frac{\ln (1 + x)}{\frac{1}{\cot (x)}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (1 + x)]}{\frac{d}{d x} [\frac{1}{\cot (x)}]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

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

Step 5.3.2

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 $g (x) = 1 + x$ .

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Step 5.3.2.1

To apply the Chain Rule, set $u$ as $1 + x$ .

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

Step 5.3.2.2

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

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

Step 5.3.2.3

Replace all occurrences of $u$ with $1 + x$ .

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 5.3.5

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

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

Step 5.3.6

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

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

Step 5.3.7

Multiply $\frac{1}{1 + x}$ by $1$ .

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

Step 5.3.8

Reorder terms.

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

Step 5.3.9

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 5.3.10

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

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

Step 5.3.11

Write $1$ as a fraction with denominator $1$ .

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

Step 5.3.12

Simplify.

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Step 5.3.12.1

Rewrite the expression.

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

Step 5.3.12.2

Multiply $\frac{\sin (x)}{\cos (x)}$ by $1$ .

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

Step 5.3.13

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 5.3.14

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

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

Step 5.3.15

Raise $\cos (x)$ to the power of $1$ .

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

Step 5.3.16

Raise $\cos (x)$ to the power of $1$ .

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

Step 5.3.17

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3.18

Add $1$ and $1$ .

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

Step 5.3.19

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

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

Step 5.3.20

Multiply $- 1$ by $- 1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x + 1}}{\frac{\cos^{2} (x) + 1 \sin (x) \sin (x)}{\cos^{2} (x)}}}$

Step 5.3.21

Multiply $\sin (x)$ by $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x + 1}}{\frac{\cos^{2} (x) + \sin (x) \sin (x)}{\cos^{2} (x)}}}$

Step 5.3.22

Raise $\sin (x)$ to the power of $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x + 1}}{\frac{\cos^{2} (x) + \sin^{1} (x) \sin (x)}{\cos^{2} (x)}}}$

Step 5.3.23

Raise $\sin (x)$ to the power of $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x + 1}}{\frac{\cos^{2} (x) + \sin^{1} (x) \sin^{1} (x)}{\cos^{2} (x)}}}$

Step 5.3.24

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x + 1}}{\frac{\cos^{2} (x) + {\sin (x)}^{1 + 1}}{\cos^{2} (x)}}}$

Step 5.3.25

Add $1$ and $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x + 1}}{\frac{\cos^{2} (x) + \sin^{2} (x)}{\cos^{2} (x)}}}$

Step 5.3.26

Simplify the numerator.

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Step 5.3.26.1

Rearrange terms.

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{x + 1}}{\frac{\sin^{2} (x) + \cos^{2} (x)}{\cos^{2} (x)}}}$

Step 5.3.26.2

Apply pythagorean identity.

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

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.5

Combine $\frac{1}{x + 1}$ and $\cos^{2} (x)$ .

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

$e^{\frac{\cos^{2} (0)}{0 + 1}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$e^{\frac{1^{2}}{0 + 1}}$

Step 8.1.2

One to any power is one.

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

Step 8.2

Add $0$ and $1$ .

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

Step 8.3

Divide $1$ by $1$ .

$e^{1}$

Step 9

Simplify.

$e$