Calculus Examples

$lim_{x \to \frac{π}{2}} \frac{\tan (4 x)}{\tan (5 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (5 x)$ in terms of sines and cosines.

$lim_{x \to \frac{π}{2}} \frac{\tan (4 x)}{\frac{\sin (5 x)}{\cos (5 x)}}$

Step 1.2

Rewrite $\tan (4 x)$ in terms of sines and cosines.

$lim_{x \to \frac{π}{2}} \frac{\frac{\sin (4 x)}{\cos (4 x)}}{\frac{\sin (5 x)}{\cos (5 x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (5 x)}{\cos (5 x)}$ .

$lim_{x \to \frac{π}{2}} \frac{\sin (4 x)}{\cos (4 x)} \cdot \frac{\cos (5 x)}{\sin (5 x)}$

Step 1.4

Simplify.

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Step 1.4.1

Convert from $\frac{\sin (4 x)}{\cos (4 x)}$ to $\tan (4 x)$ .

$lim_{x \to \frac{π}{2}} \tan (4 x) \frac{\cos (5 x)}{\sin (5 x)}$

Step 1.4.2

Convert from $\frac{\cos (5 x)}{\sin (5 x)}$ to $\cot (5 x)$ .

$lim_{x \to \frac{π}{2}} \tan (4 x) \cot (5 x)$

Step 2

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

$lim_{x \to \frac{π}{2}} \tan (4 x) \cdot lim_{x \to \frac{π}{2}} \cot (5 x)$

Step 3

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

$\tan (lim_{x \to \frac{π}{2}} 4 x) \cdot lim_{x \to \frac{π}{2}} \cot (5 x)$

Step 4

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

$\tan (4 lim_{x \to \frac{π}{2}} x) \cdot lim_{x \to \frac{π}{2}} \cot (5 x)$

Step 5

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

$\tan (4 lim_{x \to \frac{π}{2}} x) \cdot \cot (lim_{x \to \frac{π}{2}} 5 x)$

Step 6

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\tan (4 lim_{x \to \frac{π}{2}} x) \cdot \cot (5 lim_{x \to \frac{π}{2}} x)$

Step 7

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

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Step 7.1

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

$\tan (4 \frac{π}{2}) \cdot \cot (5 lim_{x \to \frac{π}{2}} x)$

Step 7.2

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

$\tan (4 \frac{π}{2}) \cdot \cot (5 \frac{π}{2})$

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Factor $2$ out of $4$ .

$\tan (2 (2) \frac{π}{2}) \cdot \cot (5 \frac{π}{2})$

Step 8.1.2

Cancel the common factor.

$\tan (2 \cdot 2 \frac{π}{2}) \cdot \cot (5 \frac{π}{2})$

Step 8.1.3

Rewrite the expression.

$\tan (2 π) \cdot \cot (5 \frac{π}{2})$

Step 8.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\tan (0) \cdot \cot (5 \frac{π}{2})$

Step 8.3

The exact value of $\tan (0)$ is $0$ .

$0 \cdot \cot (5 \frac{π}{2})$

Step 8.4

Combine $5$ and $\frac{π}{2}$ .

$0 \cdot \cot (\frac{5 π}{2})$

Step 8.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$0 \cdot \cot (\frac{π}{2})$

Step 8.6

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

$0 \cdot 0$

Step 8.7

Multiply $0$ by $0$ .

$0$