Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 1.2

Evaluate the limit of $4$ which is constant as $x$ approaches $- 2$ .

$\frac{lim_{x \to - 2} x - 1 \cdot 4}{2 - x - x^{2}}$

Step 2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

$\frac{- 2 - 1 \cdot 4}{2 - x - x^{2}}$

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 $4$ .

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

Step 3.1.2

Subtract $4$ from $- 2$ .

$\frac{- 6}{2 - x - x^{2}}$

Step 3.2

Factor by grouping.

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Step 3.2.1

Reorder terms.

$\frac{- 6}{- x^{2} - x + 2}$

Step 3.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 2 = - 2$ and whose sum is $b = - 1$ .

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Step 3.2.2.1

Factor $- 1$ out of $- x$ .

$\frac{- 6}{- x^{2} - (x) + 2}$

Step 3.2.2.2

Rewrite $- 1$ as $1$ plus $- 2$

$\frac{- 6}{- x^{2} + (1 - 2) x + 2}$

Step 3.2.2.3

Apply the distributive property.

$\frac{- 6}{- x^{2} + 1 x - 2 x + 2}$

Step 3.2.2.4

Multiply $x$ by $1$ .

$\frac{- 6}{- x^{2} + x - 2 x + 2}$

Step 3.2.3

Factor out the greatest common factor from each group.

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Step 3.2.3.1

Group the first two terms and the last two terms.

$\frac{- 6}{(- x^{2} + x) - 2 x + 2}$

Step 3.2.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{- 6}{x (- x + 1) + 2 (- x + 1)}$

Step 3.2.4

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$\frac{- 6}{(- x + 1) (x + 2)}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{6}{(- x + 1) (x + 2)}$

Step 3.4

Factor $- 1$ out of $- x$ .

$- \frac{6}{(- (x) + 1) (x + 2)}$

Step 3.5

Rewrite $1$ as $- 1 (- 1)$ .

$- \frac{6}{(- (x) - 1 (- 1)) (x + 2)}$

Step 3.6

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$- \frac{6}{- (x - 1) (x + 2)}$

Step 3.7

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$- \frac{6}{- 1 (x - 1) (x + 2)}$

Step 3.8

Move the negative in front of the fraction.

$- - \frac{6}{(x - 1) (x + 2)}$

Step 3.9

Multiply $- 1$ by $- 1$ .

$1 \frac{6}{(x - 1) (x + 2)}$

Step 3.10

Multiply $\frac{6}{(x - 1) (x + 2)}$ by $1$ .

$\frac{6}{(x - 1) (x + 2)}$