Calculus Examples

$lim_{θ \to \frac{π}{2}} \tan (θ^{\cos (θ)})$

Step 1

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

$\tan (lim_{θ \to \frac{π}{2}} θ^{\cos (θ)})$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $θ^{\cos (θ)}$ as $e^{\ln (θ^{\cos (θ)})}$ .

$\tan (lim_{θ \to \frac{π}{2}} e^{\ln (θ^{\cos (θ)})})$

Step 2.2

Expand $\ln (θ^{\cos (θ)})$ by moving $\cos (θ)$ outside the logarithm.

$\tan (lim_{θ \to \frac{π}{2}} e^{\cos (θ) \ln (θ)})$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$\tan (e^{lim_{θ \to \frac{π}{2}} \cos (θ) \ln (θ)})$

Step 3.2

Split the limit using the Product of Limits Rule on the limit as $θ$ approaches $\frac{π}{2}$ .

$\tan (e^{lim_{θ \to \frac{π}{2}} \cos (θ) \cdot lim_{θ \to \frac{π}{2}} \ln (θ)})$

Step 3.3

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

$\tan (e^{\cos (lim_{θ \to \frac{π}{2}} θ) \cdot lim_{θ \to \frac{π}{2}} \ln (θ)})$

Step 3.4

Move the limit inside the logarithm.

$\tan (e^{\cos (lim_{θ \to \frac{π}{2}} θ) \cdot \ln (lim_{θ \to \frac{π}{2}} θ)})$

Step 4

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $θ$ .

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Step 4.1

Evaluate the limit of $θ$ by plugging in $\frac{π}{2}$ for $θ$ .

$\tan (e^{\cos (\frac{π}{2}) \cdot \ln (lim_{θ \to \frac{π}{2}} θ)})$

Step 4.2

Evaluate the limit of $θ$ by plugging in $\frac{π}{2}$ for $θ$ .

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

Step 5

Simplify the answer.

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Step 5.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\tan (e^{0 \cdot \ln (\frac{π}{2})})$

Step 5.2

Simplify $0 \cdot \ln (\frac{π}{2})$ by moving $0$ inside the logarithm.

$\tan (e^{\ln ({(\frac{π}{2})}^{0})})$

Step 5.3

Exponentiation and log are inverse functions.

$\tan ({(\frac{π}{2})}^{0})$

Step 5.4

Apply the product rule to $\frac{π}{2}$ .

$\tan (\frac{π^{0}}{2^{0}})$

Step 5.5

Anything raised to $0$ is $1$ .

$\tan (\frac{1}{2^{0}})$

Step 5.6

Anything raised to $0$ is $1$ .

$\tan (\frac{1}{1})$

Step 5.7

Divide $1$ by $1$ .

$\tan (1)$

Step 5.8

Evaluate $\tan (1)$ .

$1.55740772$