Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{1 - 1 \cdot 1}{\cos (x)}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{\cos (x)}$

Step 3.1.3

Subtract $1$ from $1$ .

$\frac{0}{\cos (x)}$

Step 3.2

Separate fractions.

$\frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 3.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{0}{1} \sec (x)$

Step 3.4

Divide $0$ by $1$ .

$0 \sec (x)$

Step 3.5

Multiply $0$ by $\sec (x)$ .

$0$