Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $π$ .

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

Step 1.3

Evaluate the limit of $π$ which is constant as $x$ approaches $π$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Rewrite $\frac{\sin (π + π)}{\cos (x - \frac{π}{2})}$ as a product.

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

Step 3.2

Write $\sin (π + π)$ as a fraction with denominator $1$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Divide $\sin (π + π)$ by $1$ .

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

Step 3.3.2

Convert from $\frac{1}{\cos (x - \frac{π}{2})}$ to $\sec (x - \frac{π}{2})$ .

$\sin (π + π) \sec (x - \frac{π}{2})$

Step 3.4

Add $π$ and $π$ .

$\sin (2 π) \sec (x - \frac{π}{2})$

Step 3.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (0) \sec (x - \frac{π}{2})$

Step 3.6

The exact value of $\sin (0)$ is $0$ .

$0 \sec (x - \frac{π}{2})$

Step 3.7

Multiply $0$ by $\sec (x - \frac{π}{2})$ .

$0$