Calculus Examples

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

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.

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

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 $x$ approaches $0$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.2

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

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

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 $1$ .

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

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

Step 1.1.3

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

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $\cos (x) - 1$ with respect to $x$ is $\frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 1]$ .

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

Step 1.3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

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

Step 1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.3.5

Add $- \sin (x)$ and $0$ .

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

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

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

Step 1.4

Divide $- \sin (x)$ by $1$ .

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

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 $x$ .

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

Step 2.2

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

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

Step 3

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

$- \sin (0)$

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 $0$ .

$0$