Calculus Examples

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

Step 1

Apply trigonometric identities.

Tap for more steps...

Step 1.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$lim_{x \to \frac{π}{4}} \frac{\tan (2 x)}{\frac{1}{\cos (2 x)}}$

Step 1.2

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$lim_{x \to \frac{π}{4}} \frac{\frac{\sin (2 x)}{\cos (2 x)}}{\frac{1}{\cos (2 x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (2 x)}$ .

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

Step 1.4

Write $\cos (2 x)$ as a fraction with denominator $1$ .

$lim_{x \to \frac{π}{4}} \frac{\sin (2 x)}{\cos (2 x)} \cdot \frac{\cos (2 x)}{1}$

Step 1.5

Cancel the common factor of $\cos (2 x)$ .

Tap for more steps...

Step 1.5.1

Cancel the common factor.

$lim_{x \to \frac{π}{4}} \frac{\sin (2 x)}{\cos (2 x)} \cdot \frac{\cos (2 x)}{1}$

Step 1.5.2

Rewrite the expression.

$lim_{x \to \frac{π}{4}} \sin (2 x)$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

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

$\sin (lim_{x \to \frac{π}{4}} 2 x)$

Step 2.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\sin (2 lim_{x \to \frac{π}{4}} x)$

Step 3

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

$\sin (2 \frac{π}{4})$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.1

Factor $2$ out of $4$ .

$\sin (2 \frac{π}{2 (2)})$

Step 4.1.2

Cancel the common factor.

$\sin (2 \frac{π}{2 \cdot 2})$

Step 4.1.3

Rewrite the expression.

$\sin (\frac{π}{2})$

Step 4.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$1$