Calculus Examples

$lim_{x \to (\frac{π}{4})} {\tan (x)}^{\tan (2 x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${\tan (x)}^{\tan (2 x)}$ as $e^{\ln ({\tan (x)}^{\tan (2 x)})}$ .

$lim_{x \to (\frac{π}{4})} e^{\ln ({\tan (x)}^{\tan (2 x)})}$

Step 1.2

Expand $\ln ({\tan (x)}^{\tan (2 x)})$ by moving $\tan (2 x)$ outside the logarithm.

$lim_{x \to (\frac{π}{4})} e^{\tan (2 x) \ln (\tan (x))}$

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to {(\frac{π}{4})}^{-}} e^{\tan (2 x) \ln (\tan (x))}$

Step 3

Evaluate the limits by plugging in the value for the variable.

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Step 3.1

Evaluate the limit of $e^{\tan (2 x) \ln (\tan (x))}$ by plugging in $(\frac{π}{4})$ for $x$ .

$e^{\tan (2 (\frac{π}{4})) \ln (\tan ((\frac{π}{4})))}$

Step 3.2

Multiply $2$ by each element of the matrix.

$e^{\tan ((2 \frac{π}{4})) \ln (\tan ((\frac{π}{4})))}$

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Factor $2$ out of $4$ .

$e^{\tan ((2 \frac{π}{2 (2)})) \ln (\tan ((\frac{π}{4})))}$

Step 3.3.2

Cancel the common factor.

$e^{\tan ((2 \frac{π}{2 \cdot 2})) \ln (\tan ((\frac{π}{4})))}$

Step 3.3.3

Rewrite the expression.

$e^{\tan ((\frac{π}{2})) \ln (\tan ((\frac{π}{4})))}$

Step 3.4

Remove parentheses.

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Step 3.4.1

Factor $2$ out of $4$ .

$e^{\tan (2 \frac{π}{2 (2)}) \ln (\tan ((\frac{π}{4})))}$

Step 3.4.2

Cancel the common factor.

$e^{\tan (2 \frac{π}{2 \cdot 2}) \ln (\tan ((\frac{π}{4})))}$

Step 3.4.3

Rewrite the expression.

$e^{\tan (\frac{π}{2}) \ln (\tan ((\frac{π}{4})))}$

Step 3.5

The exact value of $\tan (\frac{π}{2})$ is $\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}$ .

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Step 3.5.1

Split $\frac{π}{2}$ into two angles where the values of the six trigonometric functions are known.

$e^{\tan (\frac{π}{6} + \frac{π}{3}) \ln (\tan (\frac{π}{4}))}$

Step 3.5.2

Apply the sum of angles identity.

$e^{\frac{\tan (\frac{π}{6}) + \tan (\frac{π}{3})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 3.5.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{3})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 3.5.4

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 3.5.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 3.5.6

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \sqrt{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7

Simplify $\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \sqrt{3}}$ .

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Step 3.5.7.1

Simplify each term.

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Step 3.5.7.1.1

Multiply $- \frac{\sqrt{3}}{3} \sqrt{3}$ .

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Step 3.5.7.1.1.1

Combine $\sqrt{3}$ and $\frac{\sqrt{3}}{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3} \sqrt{3}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} \sqrt{3}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.1.4

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

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1 + 1}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.1.5

Add $1$ and $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{2}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.5.7.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{1}{2} \cdot 2}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.2.4

Cancel the common factor of $2$ .

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Step 3.5.7.1.2.4.1

Cancel the common factor.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.2.4.2

Rewrite the expression.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{1}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.2.5

Evaluate the exponent.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.3

Cancel the common factor of $3$ .

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Step 3.5.7.1.3.1

Cancel the common factor.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.3.2

Rewrite the expression.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1 \cdot 1} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.1.4

Multiply $- 1$ by $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.2

Subtract $1$ from $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$

Step 3.5.7.3

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

Undefined

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$

Step 3.5.8

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

Undefined

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$

Step 3.6

Since $e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$ is undefined, the limit does not exist.

$Does not exist$

Step 4

Set up the limit as a right-sided limit.

$lim_{x \to {(\frac{π}{4})}^{+}} e^{\tan (2 x) \ln (\tan (x))}$

Step 5

Evaluate the limits by plugging in the value for the variable.

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Step 5.1

Evaluate the limit of $e^{\tan (2 x) \ln (\tan (x))}$ by plugging in $(\frac{π}{4})$ for $x$ .

$e^{\tan (2 (\frac{π}{4})) \ln (\tan ((\frac{π}{4})))}$

Step 5.2

Multiply $2$ by each element of the matrix.

$e^{\tan ((2 \frac{π}{4})) \ln (\tan ((\frac{π}{4})))}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Factor $2$ out of $4$ .

$e^{\tan ((2 \frac{π}{2 (2)})) \ln (\tan ((\frac{π}{4})))}$

Step 5.3.2

Cancel the common factor.

$e^{\tan ((2 \frac{π}{2 \cdot 2})) \ln (\tan ((\frac{π}{4})))}$

Step 5.3.3

Rewrite the expression.

$e^{\tan ((\frac{π}{2})) \ln (\tan ((\frac{π}{4})))}$

Step 5.4

Remove parentheses.

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Step 5.4.1

Factor $2$ out of $4$ .

$e^{\tan (2 \frac{π}{2 (2)}) \ln (\tan ((\frac{π}{4})))}$

Step 5.4.2

Cancel the common factor.

$e^{\tan (2 \frac{π}{2 \cdot 2}) \ln (\tan ((\frac{π}{4})))}$

Step 5.4.3

Rewrite the expression.

$e^{\tan (\frac{π}{2}) \ln (\tan ((\frac{π}{4})))}$

Step 5.5

The exact value of $\tan (\frac{π}{2})$ is $\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}$ .

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Step 5.5.1

Split $\frac{π}{2}$ into two angles where the values of the six trigonometric functions are known.

$e^{\tan (\frac{π}{6} + \frac{π}{3}) \ln (\tan (\frac{π}{4}))}$

Step 5.5.2

Apply the sum of angles identity.

$e^{\frac{\tan (\frac{π}{6}) + \tan (\frac{π}{3})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 5.5.3

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \tan (\frac{π}{3})}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 5.5.4

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \tan (\frac{π}{6}) \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 5.5.5

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \tan (\frac{π}{3})} \ln (\tan (\frac{π}{4}))}$

Step 5.5.6

The exact value of $\tan (\frac{π}{3})$ is $\sqrt{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \sqrt{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7

Simplify $\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \sqrt{3}}$ .

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Step 5.5.7.1

Simplify each term.

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Step 5.5.7.1.1

Multiply $- \frac{\sqrt{3}}{3} \sqrt{3}$ .

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Step 5.5.7.1.1.1

Combine $\sqrt{3}$ and $\frac{\sqrt{3}}{3}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3} \sqrt{3}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} \sqrt{3}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.1.4

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

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1 + 1}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.1.5

Add $1$ and $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{2}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.5.7.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{1}{2} \cdot 2}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.2.4

Cancel the common factor of $2$ .

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Step 5.5.7.1.2.4.1

Cancel the common factor.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.2.4.2

Rewrite the expression.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{1}}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.2.5

Evaluate the exponent.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.3

Cancel the common factor of $3$ .

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Step 5.5.7.1.3.1

Cancel the common factor.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.3.2

Rewrite the expression.

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1 \cdot 1} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.1.4

Multiply $- 1$ by $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.2

Subtract $1$ from $1$ .

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$

Step 5.5.7.3

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

Undefined

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$

Step 5.5.8

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

Undefined

$e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$

Step 5.6

Since $e^{\frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0} \ln (\tan (\frac{π}{4}))}$ is undefined, the limit does not exist.

$Does not exist$

Step 6

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$