Calculus Examples

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

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{2}$ .

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

Step 1.3

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

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

Step 2

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

$\frac{1 - \sin (\frac{π}{2})}{x - \frac{π}{2}}$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

Tap for more steps...

Step 3.1.1

Multiply $\frac{1 - \sin (\frac{π}{2})}{x - \frac{π}{2}}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{1 - \sin (\frac{π}{2})}{x - \frac{π}{2}}$

Step 3.1.2

Combine.

$\frac{2 (1 - \sin (\frac{π}{2}))}{2 (x - \frac{π}{2})}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 1 + 2 (- \sin (\frac{π}{2}))}{2 x + 2 (- \frac{π}{2})}$

Step 3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$\frac{2 \cdot 1 + 2 (- \sin (\frac{π}{2}))}{2 x + 2 \frac{- π}{2}}$

Step 3.3.2

Cancel the common factor.

$\frac{2 \cdot 1 + 2 (- \sin (\frac{π}{2}))}{2 x + 2 \frac{- π}{2}}$

Step 3.3.3

Rewrite the expression.

$\frac{2 \cdot 1 + 2 (- \sin (\frac{π}{2}))}{2 x - π}$

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Factor $2$ out of $2 \cdot 1 + 2 (- \sin (\frac{π}{2}))$ .

$\frac{2 (1 - \sin (\frac{π}{2}))}{2 x - π}$

Step 3.4.2

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

$\frac{2 (1 - 1 \cdot 1)}{2 x - π}$

Step 3.4.3

Multiply $- 1$ by $1$ .

$\frac{2 (1 - 1)}{2 x - π}$

Step 3.4.4

Subtract $1$ from $1$ .

$\frac{2 \cdot 0}{2 x - π}$

Step 3.5

Multiply $2$ by $0$ .

$\frac{0}{2 x - π}$

Step 3.6

Divide $0$ by $2 x - π$ .

$0$