Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Move the term $\frac{5}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.1.1.1

Cancel the common factor.

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

Step 7.1.1.2

Rewrite the expression.

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

Step 7.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin (0) + \tan (\frac{5}{2} \cdot \frac{π}{2})$

Step 7.1.3

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

$0 + \tan (\frac{5}{2} \cdot \frac{π}{2})$

Step 7.1.4

Multiply $\frac{5}{2} \cdot \frac{π}{2}$ .

Tap for more steps...

Step 7.1.4.1

Multiply $\frac{5}{2}$ by $\frac{π}{2}$ .

$0 + \tan (\frac{5 π}{2 \cdot 2})$

Step 7.1.4.2

Multiply $2$ by $2$ .

$0 + \tan (\frac{5 π}{4})$

Step 7.1.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$0 + \tan (\frac{π}{4})$

Step 7.1.6

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

$0 + 1$

Step 7.2

Add $0$ and $1$ .

$1$