Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Move the limit under the radical sign.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

Tap for more steps...

Step 10.1

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

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

Step 10.2

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

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

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Multiply the numerator and denominator of the fraction by $\sqrt{\sin (\frac{π}{2})} \cdot 2$ .

Tap for more steps...

Step 11.1.1

Multiply $\frac{\frac{1}{\sqrt{\sin (\frac{π}{2})}} - 1 \cdot 1}{\frac{π}{2} + \frac{π}{2}}$ by $\frac{\sqrt{\sin (\frac{π}{2})} \cdot 2}{\sqrt{\sin (\frac{π}{2})} \cdot 2}$ .

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

Step 11.1.2

Combine.

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Simplify by cancelling.

Tap for more steps...

Step 11.3.1

Cancel the common factor of $\sqrt{\sin (\frac{π}{2})}$ .

Tap for more steps...

Step 11.3.1.1

Factor $\sqrt{\sin (\frac{π}{2})}$ out of $\sqrt{\sin (\frac{π}{2})} \cdot 2$ .

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

Step 11.3.1.2

Cancel the common factor.

Step 11.3.1.3

Rewrite the expression.

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

Step 11.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.3.2.1

Factor $2$ out of $\sqrt{\sin (\frac{π}{2})} \cdot 2$ .

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

Step 11.3.2.2

Cancel the common factor.

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

Step 11.3.2.3

Rewrite the expression.

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

Step 11.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.3.3.1

Factor $2$ out of $\sqrt{\sin (\frac{π}{2})} \cdot 2$ .

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

Step 11.3.3.2

Cancel the common factor.

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

Step 11.3.3.3

Rewrite the expression.

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

Step 11.4

Simplify the numerator.

Tap for more steps...

Step 11.4.1

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

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

Step 11.4.2

Any root of $1$ is $1$ .

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

Step 11.4.3

Multiply $1 \cdot 2 (- 1 \cdot 1)$ .

Tap for more steps...

Step 11.4.3.1

Multiply $2$ by $1$ .

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

Step 11.4.3.2

Multiply $- 1$ by $1$ .

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

Step 11.4.3.3

Multiply $2$ by $- 1$ .

$\frac{2 - 2}{\sqrt{\sin (\frac{π}{2})} π + \sqrt{\sin (\frac{π}{2})} π}$

Step 11.4.4

Subtract $2$ from $2$ .

$\frac{0}{\sqrt{\sin (\frac{π}{2})} π + \sqrt{\sin (\frac{π}{2})} π}$

Step 11.5

Simplify the denominator.

Tap for more steps...

Step 11.5.1

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

$\frac{0}{\sqrt{1} π + \sqrt{\sin (\frac{π}{2})} π}$

Step 11.5.2

Any root of $1$ is $1$ .

$\frac{0}{1 π + \sqrt{\sin (\frac{π}{2})} π}$

Step 11.5.3

Multiply $π$ by $1$ .

$\frac{0}{π + \sqrt{\sin (\frac{π}{2})} π}$

Step 11.5.4

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

$\frac{0}{π + \sqrt{1} π}$

Step 11.5.5

Any root of $1$ is $1$ .

$\frac{0}{π + 1 π}$

Step 11.5.6

Multiply $π$ by $1$ .

$\frac{0}{π + π}$

Step 11.5.7

Add $π$ and $π$ .

$\frac{0}{2 π}$

Step 11.6

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 11.6.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 π}$

Step 11.6.2

Cancel the common factors.

Tap for more steps...

Step 11.6.2.1

Factor $2$ out of $2 π$ .

$\frac{2 (0)}{2 (π)}$

Step 11.6.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 π}$

Step 11.6.2.3

Rewrite the expression.

$\frac{0}{π}$

Step 11.7

Divide $0$ by $π$ .

$0$