Calculus Examples

$\frac{lim_{x \to 4} 3 - x}{x^{2} - 2 x - 8}$

Step 1

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

$\frac{lim_{x \to 4} 3 - lim_{x \to 4} x}{x^{2} - 2 x - 8}$

Step 2

Evaluate the limit of $3$ which is constant as $x$ approaches $4$ .

$\frac{3 - lim_{x \to 4} x}{x^{2} - 2 x - 8}$

Step 3

Simplify terms.

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Step 3.1

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

$\frac{3 - 4}{x^{2} - 2 x - 8}$

Step 3.2

Simplify the answer.

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Step 3.2.1

Subtract $4$ from $3$ .

$\frac{- 1}{x^{2} - 2 x - 8}$

Step 3.2.2

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 3.2.2.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 3.2.2.2

Write the factored form using these integers.

$\frac{- 1}{(x - 4) (x + 2)}$

Step 3.2.3

Move the negative in front of the fraction.

$- \frac{1}{(x - 4) (x + 2)}$