Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

$\frac{0}{x - \frac{π}{2}}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{0}{x \cdot \frac{2}{2} - \frac{π}{2}}$

Step 3.2.2

Combine $x$ and $\frac{2}{2}$ .

$\frac{0}{\frac{x \cdot 2}{2} - \frac{π}{2}}$

Step 3.2.3

Combine the numerators over the common denominator.

$\frac{0}{\frac{x \cdot 2 - π}{2}}$

Step 3.2.4

Move $2$ to the left of $x$ .

$\frac{0}{\frac{2 x - π}{2}}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$0 \frac{2}{2 x - π}$

Step 3.4

Multiply $0$ by $\frac{2}{2 x - π}$ .

$0$