Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 2

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

$\sin (\frac{1}{2} π - π)$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{2}$ and $π$ .

$\sin (\frac{π}{2} - π)$

Step 3.2

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

$\sin (\frac{π}{2} - π \cdot \frac{2}{2})$

Step 3.3

Combine $- π$ and $\frac{2}{2}$ .

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

Step 3.4

Combine the numerators over the common denominator.

$\sin (\frac{π - π \cdot 2}{2})$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $- 1$ .

$\sin (\frac{π - 2 π}{2})$

Step 3.5.2

Subtract $2 π$ from $π$ .

$\sin (\frac{- π}{2})$

Step 3.6

Move the negative in front of the fraction.

$\sin (- \frac{π}{2})$

Step 3.7

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

$\sin (\frac{3 π}{2})$

Step 3.8

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \sin (\frac{π}{2})$

Step 3.9

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

$- 1 \cdot 1$

Step 3.10

Multiply $- 1$ by $1$ .

$- 1$