Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as

$h$ approaches

$0$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

Evaluate the limit of

$1$ which is constant as

$h$ approaches

$0$ .

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

Step 1.1.2.2

Evaluate the limit of

$h$ by plugging in

$0$ for

$h$ .

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

The exact value of

$\cos (0)$ is

$1$ .

$\frac{1 - 1 \cdot 1}{lim_{h \to 0} h}$

Step 1.1.2.3.1.2

Multiply

$- 1$ by

$1$ .

$\frac{1 - 1}{lim_{h \to 0} h}$

Step 1.1.2.3.2

Subtract

$1$ from

$1$ .

$\frac{0}{lim_{h \to 0} h}$

Step 1.1.3

Evaluate the limit of

$h$ by plugging in

$0$ for

$h$ .

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since

$\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{h \to 0} \frac{\cos (h) - 1}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [\cos (h) - 1]}{\frac{d}{d h} [h]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [\cos (h) - 1]}{\frac{d}{d h} [h]}$

Step 1.3.2

By the Sum Rule, the derivative of

$\cos (h) - 1$ with respect to

$h$ is

$\frac{d}{d h} [\cos (h)] + \frac{d}{d h} [- 1]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [\cos (h)] + \frac{d}{d h} [- 1]}{\frac{d}{d h} [h]}$

Step 1.3.3

The derivative of

$\cos (h)$ with respect to

$h$ is

$- \sin (h)$ .

$lim_{h \to 0} \frac{- \sin (h) + \frac{d}{d h} [- 1]}{\frac{d}{d h} [h]}$

Step 1.3.4

Since

$- 1$ is constant with respect to

$h$ , the derivative of

$- 1$ with respect to

$h$ is

$0$ .

$lim_{h \to 0} \frac{- \sin (h) + 0}{\frac{d}{d h} [h]}$

Step 1.3.5

Add

$- \sin (h)$ and

$0$ .

$lim_{h \to 0} \frac{- \sin (h)}{\frac{d}{d h} [h]}$

Step 1.3.6

Differentiate using the Power Rule which states that

$\frac{d}{d h} [h^{n}]$ is

$n h^{n - 1}$ where

$n = 1$ .

$lim_{h \to 0} \frac{- \sin (h)}{1}$

Step 1.4

Divide

$- \sin (h)$ by

$1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term

$- 1$ outside of the limit because it is constant with respect to

$h$ .

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

Step 2.2

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

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

Step 3

Evaluate the limit of

$h$ by plugging in

$0$ for

$h$ .

$- \sin (0)$

Step 4

Simplify the answer.

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Step 4.1

The exact value of

$\sin (0)$ is

$0$ .

$- 0$

Step 4.2

Multiply

$- 1$ by

$0$ .

$0$