Calculus Examples

$lim_{x \to 4} \frac{\sin (x - 4)}{x - 4}$

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.

$\frac{lim_{x \to 4} \sin (x - 4)}{lim_{x \to 4} x - 4}$

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

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

$\frac{\sin (lim_{x \to 4} x - 4)}{lim_{x \to 4} x - 4}$

Step 1.1.2.1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{\sin (lim_{x \to 4} x - lim_{x \to 4} 4)}{lim_{x \to 4} x - 4}$

Step 1.1.2.1.3

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\frac{\sin (lim_{x \to 4} x - 1 \cdot 4)}{lim_{x \to 4} x - 4}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{\sin (4 - 1 \cdot 4)}{lim_{x \to 4} x - 4}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $4$ .

$\frac{\sin (4 - 4)}{lim_{x \to 4} x - 4}$

Step 1.1.2.3.2

Subtract $4$ from $4$ .

$\frac{\sin (0)}{lim_{x \to 4} x - 4}$

Step 1.1.2.3.3

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

$\frac{0}{lim_{x \to 4} x - 4}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{0}{lim_{x \to 4} x - lim_{x \to 4} 4}$

Step 1.1.3.1.2

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\frac{0}{lim_{x \to 4} x - 1 \cdot 4}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{4 - 1 \cdot 4}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $4$ .

$\frac{0}{4 - 4}$

Step 1.1.3.3.2

Subtract $4$ from $4$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 4} \frac{\sin (x - 4)}{x - 4} = lim_{x \to 4} \frac{\frac{d}{d x} [\sin (x - 4)]}{\frac{d}{d x} [x - 4]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [\sin (x - 4)]}{\frac{d}{d x} [x - 4]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x - 4$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $x - 4$ .

$lim_{x \to 4} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x - 4]}{\frac{d}{d x} [x - 4]}$

Step 1.3.2.2

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

$lim_{x \to 4} \frac{\cos (u) \frac{d}{d x} [x - 4]}{\frac{d}{d x} [x - 4]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $x - 4$ .

$lim_{x \to 4} \frac{\cos (x - 4) \frac{d}{d x} [x - 4]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 4} \frac{\cos (x - 4) (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{\frac{d}{d x} [x - 4]}$

Step 1.3.4

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

$lim_{x \to 4} \frac{\cos (x - 4) (1 + \frac{d}{d x} [- 4])}{\frac{d}{d x} [x - 4]}$

Step 1.3.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{\cos (x - 4) (1 + 0)}{\frac{d}{d x} [x - 4]}$

Step 1.3.6

Add $1$ and $0$ .

$lim_{x \to 4} \frac{\cos (x - 4) \cdot 1}{\frac{d}{d x} [x - 4]}$

Step 1.3.7

Multiply $\cos (x - 4)$ by $1$ .

$lim_{x \to 4} \frac{\cos (x - 4)}{\frac{d}{d x} [x - 4]}$

Step 1.3.8

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 4} \frac{\cos (x - 4)}{\frac{d}{d x} [x] + \frac{d}{d x} [- 4]}$

Step 1.3.9

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

$lim_{x \to 4} \frac{\cos (x - 4)}{1 + \frac{d}{d x} [- 4]}$

Step 1.3.10

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{\cos (x - 4)}{1 + 0}$

Step 1.3.11

Add $1$ and $0$ .

$lim_{x \to 4} \frac{\cos (x - 4)}{1}$

Step 1.4

Divide $\cos (x - 4)$ by $1$ .

$lim_{x \to 4} \cos (x - 4)$

Step 2

Evaluate the limit.

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Step 2.1

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

$\cos (lim_{x \to 4} x - 4)$

Step 2.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\cos (lim_{x \to 4} x - lim_{x \to 4} 4)$

Step 2.3

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\cos (lim_{x \to 4} x - 1 \cdot 4)$

Step 3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\cos (4 - 1 \cdot 4)$

Step 4

Simplify the answer.

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Step 4.1

Multiply $- 1$ by $4$ .

$\cos (4 - 4)$

Step 4.2

Subtract $4$ from $4$ .

$\cos (0)$

Step 4.3

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

$1$