Calculus Examples

$\frac{lim_{h \to 0} {(1 - \cos (h))}^{2}}{h}$

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $2$ from ${(1 - \cos (h))}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{h \to 0} 1 - \cos (h))}^{2}}{h}$

Step 1.2

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

$\frac{{(lim_{h \to 0} 1 - lim_{h \to 0} \cos (h))}^{2}}{h}$

Step 1.3

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$\frac{{(1 - lim_{h \to 0} \cos (h))}^{2}}{h}$

Step 1.4

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

$\frac{{(1 - \cos (lim_{h \to 0} h))}^{2}}{h}$

Step 2

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

$\frac{{(1 - \cos (0))}^{2}}{h}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

The exact value of $\cos (0)$ is $1$ .

$\frac{{(1 - 1 \cdot 1)}^{2}}{h}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{{(1 - 1)}^{2}}{h}$

Step 3.1.3

Subtract $1$ from $1$ .

$\frac{0^{2}}{h}$

Step 3.1.4

Raising $0$ to any positive power yields $0$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$