Calculus Examples

$\frac{lim_{x \to - 1} x^{2} - 4 x}{x^{2} - 3 x - 4}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{lim_{x \to - 1} x^{2} - lim_{x \to - 1} 4 x}{x^{2} - 3 x - 4}$

Step 2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to - 1} x)}^{2} - lim_{x \to - 1} 4 x}{x^{2} - 3 x - 4}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to - 1} x)}^{2} - 4 lim_{x \to - 1} x}{x^{2} - 3 x - 4}$

Step 4

Evaluate the limits by plugging in $- 1$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{{(- 1)}^{2} - 4 lim_{x \to - 1} x}{x^{2} - 3 x - 4}$

Step 4.2

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{{(- 1)}^{2} - 4 \cdot - 1}{x^{2} - 3 x - 4}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 1$ to the power of $2$ .

$\frac{1 - 4 \cdot - 1}{x^{2} - 3 x - 4}$

Step 5.1.2

Multiply $- 4$ by $- 1$ .

$\frac{1 + 4}{x^{2} - 3 x - 4}$

Step 5.1.3

Add $1$ and $4$ .

$\frac{5}{x^{2} - 3 x - 4}$

Step 5.2

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 5.2.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 5.2.2

Write the factored form using these integers.

$\frac{5}{(x - 4) (x + 1)}$