Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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 $x$ .

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

Step 6.2

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

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

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

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

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

Step 7.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{\csc (\frac{π}{2})}$

Step 7.1.3

Subtract $1$ from $1$ .

$\frac{0}{\csc (\frac{π}{2})}$

Step 7.2

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

$\frac{0}{1}$

Step 7.3

Divide $0$ by $1$ .

$0$