Calculus Examples

$\frac{lim_{x \to - 8} \sqrt{2 - x}}{x^{2} + x - 12}$

Step 1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to - 8} 2 - x}}{x^{2} + x - 12}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\frac{\sqrt{lim_{x \to - 8} 2 - lim_{x \to - 8} x}}{x^{2} + x - 12}$

Step 3

Evaluate the limit of $2$ which is constant as $x$ approaches $- 8$ .

$\frac{\sqrt{2 - lim_{x \to - 8} x}}{x^{2} + x - 12}$

Step 4

Simplify terms.

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Step 4.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt{2 + 8}}{x^{2} + x - 12}$

Step 4.2

Simplify the answer.

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Step 4.2.1

Add $2$ and $8$ .

$\frac{\sqrt{10}}{x^{2} + x - 12}$

Step 4.2.2

Factor $x^{2} + x - 12$ using the AC method.

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Step 4.2.2.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 4.2.2.2

Write the factored form using these integers.

$\frac{\sqrt{10}}{(x - 3) (x + 4)}$