Calculus Examples

$lim_{x \to (\frac{π}{2})} 3^{e \tan (x)}$

Step 1

Set up the limit as a left-sided limit.

$lim_{x \to {(\frac{π}{2})}^{-}} 3^{e \tan (x)}$

Step 2

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

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Step 2.1

Evaluate the limit of $3^{e \tan (x)}$ by plugging in $(\frac{π}{2})$ for $x$ .

$3^{e \tan ((\frac{π}{2}))}$

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Apply the sum of angles identity.

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \sqrt{3}}}$

Step 2.2.7

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

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Step 2.2.7.1

Simplify each term.

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Step 2.2.7.1.1

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

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Step 2.2.7.1.1.1

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3} \sqrt{3}}{3}}}$

Step 2.2.7.1.1.2

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} \sqrt{3}}{3}}}$

Step 2.2.7.1.1.3

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3}}}$

Step 2.2.7.1.1.4

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1 + 1}}{3}}}$

Step 2.2.7.1.1.5

Add $1$ and $1$ .

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{2}}{3}}}$

Step 2.2.7.1.2

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

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Step 2.2.7.1.2.1

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{3}}}$

Step 2.2.7.1.2.2

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

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

Step 2.2.7.1.2.3

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}}}$

Step 2.2.7.1.2.4

Cancel the common factor of $2$ .

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Step 2.2.7.1.2.4.1

Cancel the common factor.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}}}$

Step 2.2.7.1.2.4.2

Rewrite the expression.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{1}}{3}}}$

Step 2.2.7.1.2.5

Evaluate the exponent.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}}}$

Step 2.2.7.1.3

Cancel the common factor of $3$ .

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Step 2.2.7.1.3.1

Cancel the common factor.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}}}$

Step 2.2.7.1.3.2

Rewrite the expression.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1 \cdot 1}}$

Step 2.2.7.1.4

Multiply $- 1$ by $1$ .

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1}}$

Step 2.2.7.2

Subtract $1$ from $1$ .

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

Step 2.2.7.3

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

Undefined

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

Step 2.2.8

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

Undefined

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

Step 2.3

Since $3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}}$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

$lim_{x \to {(\frac{π}{2})}^{+}} 3^{e \tan (x)}$

Step 4

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

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Step 4.1

Evaluate the limit of $3^{e \tan (x)}$ by plugging in $(\frac{π}{2})$ for $x$ .

$3^{e \tan ((\frac{π}{2}))}$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Apply the sum of angles identity.

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

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

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

Step 4.2.6

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3}}{3} \sqrt{3}}}$

Step 4.2.7

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

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Step 4.2.7.1

Simplify each term.

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Step 4.2.7.1.1

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

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Step 4.2.7.1.1.1

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{\sqrt{3} \sqrt{3}}{3}}}$

Step 4.2.7.1.1.2

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} \sqrt{3}}{3}}}$

Step 4.2.7.1.1.3

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{3}}}$

Step 4.2.7.1.1.4

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{1 + 1}}{3}}}$

Step 4.2.7.1.1.5

Add $1$ and $1$ .

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{\sqrt{3}}^{2}}{3}}}$

Step 4.2.7.1.2

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

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Step 4.2.7.1.2.1

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{3}}}$

Step 4.2.7.1.2.2

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

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

Step 4.2.7.1.2.3

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

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}}}$

Step 4.2.7.1.2.4

Cancel the common factor of $2$ .

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Step 4.2.7.1.2.4.1

Cancel the common factor.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{\frac{2}{2}}}{3}}}$

Step 4.2.7.1.2.4.2

Rewrite the expression.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3^{1}}{3}}}$

Step 4.2.7.1.2.5

Evaluate the exponent.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}}}$

Step 4.2.7.1.3

Cancel the common factor of $3$ .

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Step 4.2.7.1.3.1

Cancel the common factor.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - \frac{3}{3}}}$

Step 4.2.7.1.3.2

Rewrite the expression.

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1 \cdot 1}}$

Step 4.2.7.1.4

Multiply $- 1$ by $1$ .

$3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{1 - 1}}$

Step 4.2.7.2

Subtract $1$ from $1$ .

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

Step 4.2.7.3

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

Undefined

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

Step 4.2.8

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

Undefined

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

Step 4.3

Since $3^{e \frac{\frac{\sqrt{3}}{3} + \sqrt{3}}{0}}$ is undefined, the limit does not exist.

$Does not exist$

Step 5

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

$Does not exist$