Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $π$ for $x$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Combine $\frac{1}{2}$ and $π$ .

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

Step 3.1.2

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

$\frac{0}{\frac{x}{2} - \frac{π}{2}}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{0}{\frac{x - π}{2}}$

Step 3.3

Multiply the numerator by the reciprocal of the denominator.

$0 \frac{2}{x - π}$

Step 3.4

Multiply $0$ by $\frac{2}{x - π}$ .

$0$