Precalculus Examples

$\frac{lim_{x \to 2} x^{4} - 16}{x^{2} + x - 6}$

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 $2$ .

$\frac{lim_{x \to 2} x^{4} - lim_{x \to 2} 16}{x^{2} + x - 6}$

Step 1.2

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 2} x)}^{4} - lim_{x \to 2} 16}{x^{2} + x - 6}$

Step 1.3

Evaluate the limit of $16$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{4} - 1 \cdot 16}{x^{2} + x - 6}$

Step 2

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

$\frac{2^{4} - 1 \cdot 16}{x^{2} + x - 6}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $2$ to the power of $4$ .

$\frac{16 - 1 \cdot 16}{x^{2} + x - 6}$

Step 3.1.2

Multiply $- 1$ by $16$ .

$\frac{16 - 16}{x^{2} + x - 6}$

Step 3.1.3

Subtract $16$ from $16$ .

$\frac{0}{x^{2} + x - 6}$

Step 3.2

Factor $x^{2} + x - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 3.2.2

Write the factored form using these integers.

$\frac{0}{(x - 2) (x + 3)}$

Step 3.3

Divide $0$ by $(x - 2) (x + 3)$ .

$0$