Calculus Examples

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

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite

{\tan (x)}^{\tan (x)}

${\tan (x)}^{\tan (x)}$ as

e^{\ln ({\tan (x)}^{\tan (x)})}

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

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

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

Step 1.2

Expand

\ln ({\tan (x)}^{\tan (x)})

$\ln ({\tan (x)}^{\tan (x)})$ by moving

\tan (x)

$\tan (x)$ outside the logarithm.

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

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

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

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

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

Step 2.2

Split the limit using the Product of Limits Rule on the limit as

x

$x$ approaches

\frac{π}{4}

$\frac{π}{4}$ .

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

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

Step 2.3

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

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

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

Step 2.4

Move the limit inside the logarithm.

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

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

Step 2.5

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

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

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

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

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

Step 3

Evaluate the limits by plugging in

\frac{π}{4}

$\frac{π}{4}$ for all occurrences of

x

$x$ .

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Step 3.1

Evaluate the limit of

x

$x$ by plugging in

\frac{π}{4}

$\frac{π}{4}$ for

x

$x$ .

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

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

Step 3.2

Evaluate the limit of

x

$x$ by plugging in

\frac{π}{4}

$\frac{π}{4}$ for

x

$x$ .

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

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

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

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

Step 4

Simplify the answer.

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Step 4.1

The exact value of

\tan (\frac{π}{4})

$\tan (\frac{π}{4})$ is

1

$1$ .

e^{1 \cdot \ln (\tan (\frac{π}{4}))}

$e^{1 \cdot \ln (\tan (\frac{π}{4}))}$

Step 4.2

Simplify

1 \cdot \ln (\tan (\frac{π}{4}))

$1 \cdot \ln (\tan (\frac{π}{4}))$ by moving

1

$1$ inside the logarithm.

e^{\ln (\tan^{1} (\frac{π}{4}))}

$e^{\ln (\tan^{1} (\frac{π}{4}))}$

Step 4.3

Exponentiation and log are inverse functions.

\tan^{1} (\frac{π}{4})

$\tan^{1} (\frac{π}{4})$

Step 4.4

The exact value of

\tan (\frac{π}{4})

$\tan (\frac{π}{4})$ is

1

$1$ .

1^{1}

$1^{1}$

Step 4.5

Evaluate the exponent.

1

$1$

1

$1$