Calculus Examples

$\frac{lim_{x \to 4} \sin (x - 4)}{x^{2} + x - 20}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{\sin (lim_{x \to 4} x - 4)}{x^{2} + x - 20}$

Step 1.2

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

$\frac{\sin (lim_{x \to 4} x - lim_{x \to 4} 4)}{x^{2} + x - 20}$

Step 1.3

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

$\frac{\sin (lim_{x \to 4} x - 1 \cdot 4)}{x^{2} + x - 20}$

Step 2

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

$\frac{\sin (4 - 1 \cdot 4)}{x^{2} + x - 20}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $- 1$ by $4$ .

$\frac{\sin (4 - 4)}{x^{2} + x - 20}$

Step 3.1.2

Subtract $4$ from $4$ .

$\frac{\sin (0)}{x^{2} + x - 20}$

Step 3.1.3

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

$\frac{0}{x^{2} + x - 20}$

Step 3.2

Factor $x^{2} + x - 20$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 3.2.2

Write the factored form using these integers.

$\frac{0}{(x - 4) (x + 5)}$

Step 3.3

Divide $0$ by $(x - 4) (x + 5)$ .

$0$