Calculus Examples

$lim_{x \to 0^{+}} {(3 x + 1)}^{\cot (x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(3 x + 1)}^{\cot (x)}$ as $e^{\ln ({(3 x + 1)}^{\cot (x)})}$ .

$lim_{x \to 0^{+}} e^{\ln ({(3 x + 1)}^{\cot (x)})}$

Step 1.2

Expand $\ln ({(3 x + 1)}^{\cot (x)})$ by moving $\cot (x)$ outside the logarithm.

$lim_{x \to 0^{+}} e^{\cot (x) \ln (3 x + 1)}$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 0^{+}} \cot (x) \ln (3 x + 1)}$

Step 3

Rewrite $\cot (x) \ln (3 x + 1)$ as $\frac{\ln (3 x + 1)}{\frac{1}{\cot (x)}}$ .

$e^{lim_{x \to 0^{+}} \frac{\ln (3 x + 1)}{\frac{1}{\cot (x)}}}$

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.

$e^{\frac{lim_{x \to 0^{+}} \ln (3 x + 1)}{lim_{x \to 0^{+}} \frac{1}{\cot (x)}}}$

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

Move the limit inside the logarithm.

$e^{\frac{\ln (lim_{x \to 0^{+}} 3 x + 1)}{lim_{x \to 0^{+}} \frac{1}{\cot (x)}}}$

Step 4.1.2.1.2

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

$e^{\frac{\ln (lim_{x \to 0^{+}} 3 x + lim_{x \to 0^{+}} 1)}{lim_{x \to 0^{+}} \frac{1}{\cot (x)}}}$

Step 4.1.2.1.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{\ln (3 lim_{x \to 0^{+}} x + lim_{x \to 0^{+}} 1)}{lim_{x \to 0^{+}} \frac{1}{\cot (x)}}}$

Step 4.1.2.1.4

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

$e^{\frac{\ln (3 lim_{x \to 0^{+}} x + 1)}{lim_{x \to 0^{+}} \frac{1}{\cot (x)}}}$

Step 4.1.2.2

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

$e^{\frac{\ln (3 \cdot 0 + 1)}{lim_{x \to 0^{+}} \frac{1}{\cot (x)}}}$

Step 4.1.2.3

Simplify the answer.

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Step 4.1.2.3.1

Multiply $3$ by $0$ .

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

Step 4.1.2.3.2

Add $0$ and $1$ .

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

Step 4.1.2.3.3

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

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

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Apply trigonometric identities.

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Step 4.1.3.1.1

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

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

Step 4.1.3.1.2

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 4.1.3.1.3

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

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

Step 4.1.3.5

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

Undefined

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

Step 4.1.4

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

Undefined

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

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.

$lim_{x \to 0^{+}} \frac{\ln (3 x + 1)}{\frac{1}{\cot (x)}} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (3 x + 1)]}{\frac{d}{d x} [\frac{1}{\cot (x)}]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\ln (3 x + 1)]}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.2

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 $g (x) = 3 x + 1$ .

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Step 4.3.2.1

To apply the Chain Rule, set $u$ as $3 x + 1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x + 1]}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.2.2

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

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{u} \frac{d}{d x} [3 x + 1]}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.2.3

Replace all occurrences of $u$ with $3 x + 1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{3 x + 1} \frac{d}{d x} [3 x + 1]}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.3

By the Sum Rule, the derivative of $3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{3 x + 1} (\frac{d}{d x} [3 x] + \frac{d}{d x} [1])}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{3 x + 1} (3 \frac{d}{d x} [x] + \frac{d}{d x} [1])}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.5

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

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{3 x + 1} (3 \cdot 1 + \frac{d}{d x} [1])}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.6

Multiply $3$ by $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{3 x + 1} (3 + \frac{d}{d x} [1])}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.7

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

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{3 x + 1} (3 + 0)}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.8

Add $3$ and $0$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{1}{3 x + 1} \cdot 3}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.9

Combine $\frac{1}{3 x + 1}$ and $3$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{d}{d x} [\frac{1}{\cot (x)}]}}$

Step 4.3.10

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{d}{d x} [\frac{1}{\frac{\cos (x)}{\sin (x)}}]}}$

Step 4.3.11

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{d}{d x} [1 \frac{\sin (x)}{\cos (x)}]}}$

Step 4.3.12

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{d}{d x} [\frac{1}{1} \cdot \frac{\sin (x)}{\cos (x)}]}}$

Step 4.3.13

Simplify.

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Step 4.3.13.1

Rewrite the expression.

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{d}{d x} [1 \frac{\sin (x)}{\cos (x)}]}}$

Step 4.3.13.2

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{d}{d x} [\frac{\sin (x)}{\cos (x)}]}}$

Step 4.3.14

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos (x) \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}}}$

Step 4.3.15

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos (x) \cos (x) - \sin (x) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}}}$

Step 4.3.16

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{1} (x) \cos (x) - \sin (x) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}}}$

Step 4.3.17

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{1} (x) \cos^{1} (x) - \sin (x) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}}}$

Step 4.3.18

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{{\cos (x)}^{1 + 1} - \sin (x) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}}}$

Step 4.3.19

Add $1$ and $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) - \sin (x) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}}}$

Step 4.3.20

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) - \sin (x) \cdot - 1 \sin (x)}{\cos^{2} (x)}}}$

Step 4.3.21

Multiply $- 1$ by $- 1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) + 1 \sin (x) \sin (x)}{\cos^{2} (x)}}}$

Step 4.3.22

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) + \sin (x) \sin (x)}{\cos^{2} (x)}}}$

Step 4.3.23

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) + \sin^{1} (x) \sin (x)}{\cos^{2} (x)}}}$

Step 4.3.24

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) + \sin^{1} (x) \sin^{1} (x)}{\cos^{2} (x)}}}$

Step 4.3.25

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

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) + {\sin (x)}^{1 + 1}}{\cos^{2} (x)}}}$

Step 4.3.26

Add $1$ and $1$ .

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\cos^{2} (x) + \sin^{2} (x)}{\cos^{2} (x)}}}$

Step 4.3.27

Simplify the numerator.

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Step 4.3.27.1

Rearrange terms.

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{\sin^{2} (x) + \cos^{2} (x)}{\cos^{2} (x)}}}$

Step 4.3.27.2

Apply pythagorean identity.

$e^{lim_{x \to 0^{+}} \frac{\frac{3}{3 x + 1}}{\frac{1}{\cos^{2} (x)}}}$

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

$e^{lim_{x \to 0^{+}} \frac{3}{3 x + 1} \cos^{2} (x)}$

Step 4.5

Combine $\frac{3}{3 x + 1}$ and $\cos^{2} (x)$ .

$e^{lim_{x \to 0^{+}} \frac{3 \cos^{2} (x)}{3 x + 1}}$

Step 5

Evaluate the limit.

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Step 5.1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$e^{3 lim_{x \to 0^{+}} \frac{\cos^{2} (x)}{3 x + 1}}$

Step 5.2

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

$e^{3 \frac{lim_{x \to 0^{+}} \cos^{2} (x)}{lim_{x \to 0^{+}} 3 x + 1}}$

Step 5.3

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

$e^{3 \frac{{(lim_{x \to 0^{+}} \cos (x))}^{2}}{lim_{x \to 0^{+}} 3 x + 1}}$

Step 5.4

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

$e^{3 \frac{\cos^{2} (lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} 3 x + 1}}$

Step 5.5

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

$e^{3 \frac{\cos^{2} (lim_{x \to 0^{+}} x)}{lim_{x \to 0^{+}} 3 x + lim_{x \to 0^{+}} 1}}$

Step 5.6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$e^{3 \frac{\cos^{2} (lim_{x \to 0^{+}} x)}{3 lim_{x \to 0^{+}} x + lim_{x \to 0^{+}} 1}}$

Step 5.7

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

$e^{3 \frac{\cos^{2} (lim_{x \to 0^{+}} x)}{3 lim_{x \to 0^{+}} x + 1}}$

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 $x$ .

$e^{3 \frac{\cos^{2} (0)}{3 lim_{x \to 0^{+}} x + 1}}$

Step 6.2

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

$e^{3 \frac{\cos^{2} (0)}{3 \cdot 0 + 1}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$e^{3 \frac{1^{2}}{3 \cdot 0 + 1}}$

Step 7.1.2

One to any power is one.

$e^{3 \frac{1}{3 \cdot 0 + 1}}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $3$ by $0$ .

$e^{3 \frac{1}{0 + 1}}$

Step 7.2.2

Add $0$ and $1$ .

$e^{3 (\frac{1}{1})}$

Step 7.3

Cancel the common factor of $1$ .

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Step 7.3.1

Cancel the common factor.

$e^{3 (\frac{1}{1})}$

Step 7.3.2

Rewrite the expression.

$e^{3 \cdot 1}$

Step 7.4

Multiply $3$ by $1$ .

$e^{3}$