Calculus Examples

$lim_{x \to 0} \frac{7 - 7 \cos (x)}{x}$

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.

$\frac{lim_{x \to 0} 7 - 7 \cos (x)}{lim_{x \to 0} x}$

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 $0$ .

$\frac{lim_{x \to 0} 7 - lim_{x \to 0} 7 \cos (x)}{lim_{x \to 0} x}$

Step 1.1.2.1.2

Evaluate the limit of $7$ which is constant as $x$ approaches $0$ .

$\frac{7 - lim_{x \to 0} 7 \cos (x)}{lim_{x \to 0} x}$

Step 1.1.2.1.3

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{7 - 7 lim_{x \to 0} \cos (x)}{lim_{x \to 0} x}$

Step 1.1.2.1.4

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

$\frac{7 - 7 \cos (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 1.1.2.2

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

$\frac{7 - 7 \cos (0)}{lim_{x \to 0} x}$

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 $1$ .

$\frac{7 - 7 \cdot 1}{lim_{x \to 0} x}$

Step 1.1.2.3.1.2

Multiply $- 7$ by $1$ .

$\frac{7 - 7}{lim_{x \to 0} x}$

Step 1.1.2.3.2

Subtract $7$ from $7$ .

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

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.

$lim_{x \to 0} \frac{7 - 7 \cos (x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [7 - 7 \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [7 - 7 \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.2

By the Sum Rule, the derivative of $7 - 7 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 7 \cos (x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [7] + \frac{d}{d x} [- 7 \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{0 + \frac{d}{d x} [- 7 \cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- 7 \cos (x)]$ .

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Step 1.3.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 \cos (x)$ with respect to $x$ is $- 7 \frac{d}{d x} [\cos (x)]$ .

$lim_{x \to 0} \frac{0 - 7 \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x]}$

Step 1.3.4.2

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

$lim_{x \to 0} \frac{0 - 7 (- \sin (x))}{\frac{d}{d x} [x]}$

Step 1.3.4.3

Multiply $- 1$ by $- 7$ .

$lim_{x \to 0} \frac{0 + 7 \sin (x)}{\frac{d}{d x} [x]}$

Step 1.3.5

Add $0$ and $7 \sin (x)$ .

$lim_{x \to 0} \frac{7 \sin (x)}{\frac{d}{d x} [x]}$

Step 1.3.6

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

$lim_{x \to 0} \frac{7 \sin (x)}{1}$

Step 1.4

Divide $7 \sin (x)$ by $1$ .

$lim_{x \to 0} 7 \sin (x)$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$7 lim_{x \to 0} \sin (x)$

Step 2.2

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

$7 \sin (lim_{x \to 0} x)$

Step 3

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

$7 \sin (0)$

Step 4

Simplify the answer.

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Step 4.1

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

$7 \cdot 0$

Step 4.2

Multiply $7$ by $0$ .

$0$