Precalculus Examples

$lim_{x \to 4} \frac{x^{4} - 25}{x^{2} - 5}$

Step 1

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

$\frac{lim_{x \to 4} x^{4} - 25}{lim_{x \to 4} x^{2} - 5}$

Step 2

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

$\frac{lim_{x \to 4} x^{4} - lim_{x \to 4} 25}{lim_{x \to 4} x^{2} - 5}$

Step 3

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

$\frac{{(lim_{x \to 4} x)}^{4} - lim_{x \to 4} 25}{lim_{x \to 4} x^{2} - 5}$

Step 4

Evaluate the limit of $25$ which is constant as $x$ approaches $4$ .

$\frac{{(lim_{x \to 4} x)}^{4} - 1 \cdot 25}{lim_{x \to 4} x^{2} - 5}$

Step 5

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

$\frac{{(lim_{x \to 4} x)}^{4} - 1 \cdot 25}{lim_{x \to 4} x^{2} - lim_{x \to 4} 5}$

Step 6

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

$\frac{{(lim_{x \to 4} x)}^{4} - 1 \cdot 25}{{(lim_{x \to 4} x)}^{2} - lim_{x \to 4} 5}$

Step 7

Evaluate the limit of $5$ which is constant as $x$ approaches $4$ .

$\frac{{(lim_{x \to 4} x)}^{4} - 1 \cdot 25}{{(lim_{x \to 4} x)}^{2} - 1 \cdot 5}$

Step 8

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 8.1

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

$\frac{4^{4} - 1 \cdot 25}{{(lim_{x \to 4} x)}^{2} - 1 \cdot 5}$

Step 8.2

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

$\frac{4^{4} - 1 \cdot 25}{4^{2} - 1 \cdot 5}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{256 - 1 \cdot 25}{4^{2} - 1 \cdot 5}$

Step 9.1.2

Multiply $- 1$ by $25$ .

$\frac{256 - 25}{4^{2} - 1 \cdot 5}$

Step 9.1.3

Subtract $25$ from $256$ .

$\frac{231}{4^{2} - 1 \cdot 5}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$\frac{231}{16 - 1 \cdot 5}$

Step 9.2.2

Multiply $- 1$ by $5$ .

$\frac{231}{16 - 5}$

Step 9.2.3

Subtract $5$ from $16$ .

$\frac{231}{11}$

Step 9.3

Divide $231$ by $11$ .

$21$