Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{\cos (\frac{π}{2})}$

Step 7.1.3

Subtract $1$ from $1$ .

$\frac{0}{\cos (\frac{π}{2})}$

Step 7.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{0}{0}$

Step 7.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 8

The expression contains a division by $0$ . The expression is undefined.

Undefined