Calculus Examples

$\frac{lim_{h \to 0} \sin (π + h) - \sin (π)}{h}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} \sin (π + h) - lim_{h \to 0} \sin (π)}{h}$

Step 1.2

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

$\frac{\sin (lim_{h \to 0} π + h) - lim_{h \to 0} \sin (π)}{h}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{\sin (lim_{h \to 0} π + lim_{h \to 0} h) - lim_{h \to 0} \sin (π)}{h}$

Step 1.4

Evaluate the limit of $π$ which is constant as $h$ approaches $0$ .

$\frac{\sin (π + lim_{h \to 0} h) - lim_{h \to 0} \sin (π)}{h}$

Step 1.5

Evaluate the limit of $\sin (π)$ which is constant as $h$ approaches $0$ .

$\frac{\sin (π + lim_{h \to 0} h) - \sin (π)}{h}$

Step 2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{\sin (π + 0) - \sin (π)}{h}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $π$ and $0$ .

$\frac{\sin (π) - \sin (π)}{h}$

Step 3.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sin (0) - \sin (π)}{h}$

Step 3.1.3

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

$\frac{0 - \sin (π)}{h}$

Step 3.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{0 - \sin (0)}{h}$

Step 3.1.5

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

$\frac{0 - 0}{h}$

Step 3.1.6

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{h}$

Step 3.1.7

Add $0$ and $0$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$