Calculus Examples

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

Step 1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{4}$ .

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

Step 3

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

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

Step 4

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

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

Step 4.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$4 \frac{\frac{π}{4}}{\tan (\frac{π}{4})}$

Step 5

Simplify the answer.

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Step 5.1

Separate fractions.

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

Step 5.2

Rewrite $\tan (\frac{π}{4})$ in terms of sines and cosines.

$4 (\frac{\frac{π}{4}}{1} \cdot \frac{1}{\frac{\sin (\frac{π}{4})}{\cos (\frac{π}{4})}})$

Step 5.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (\frac{π}{4})}{\cos (\frac{π}{4})}$ .

$4 (\frac{\frac{π}{4}}{1} \cdot \frac{\cos (\frac{π}{4})}{\sin (\frac{π}{4})})$

Step 5.4

Convert from $\frac{\cos (\frac{π}{4})}{\sin (\frac{π}{4})}$ to $\cot (\frac{π}{4})$ .

$4 (\frac{\frac{π}{4}}{1} \cot (\frac{π}{4}))$

Step 5.5

Divide $\frac{π}{4}$ by $1$ .

$4 (\frac{π}{4} \cot (\frac{π}{4}))$

Step 5.6

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$4 (\frac{π}{4} \cdot 1)$

Step 5.7

Multiply $\frac{π}{4}$ by $1$ .

$4 \frac{π}{4}$

Step 5.8

Cancel the common factor of $4$ .

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Step 5.8.1

Cancel the common factor.

$4 \frac{π}{4}$

Step 5.8.2

Rewrite the expression.

$π$

Step 6

The result can be shown in multiple forms.

Exact Form:

$π$

Decimal Form:

$3.14159265 \dots$