Calculus Examples

$\frac{lim_{x \to 5} {(x^{2} - 24)}^{4 - 1}}{x - 5}$

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $4 - 1$ from ${(x^{2} - 24)}^{4 - 1}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 5} x^{2} - 24)}^{4 - 1}}{x - 5}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{{(lim_{x \to 5} x^{2} - lim_{x \to 5} 24)}^{4 - 1}}{x - 5}$

Step 1.3

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

$\frac{{({(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 24)}^{4 - 1}}{x - 5}$

Step 1.4

Evaluate the limit of $24$ which is constant as $x$ approaches $5$ .

$\frac{{({(lim_{x \to 5} x)}^{2} - 1 \cdot 24)}^{4 - 1}}{x - 5}$

Step 2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{{(5^{2} - 1 \cdot 24)}^{4 - 1}}{x - 5}$

Step 3

Simplify the numerator.

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Step 3.1

Simplify each term.

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Step 3.1.1

Raise $5$ to the power of $2$ .

$\frac{{(25 - 1 \cdot 24)}^{4 - 1}}{x - 5}$

Step 3.1.2

Multiply $- 1$ by $24$ .

$\frac{{(25 - 24)}^{4 - 1}}{x - 5}$

Step 3.2

Subtract $24$ from $25$ .

$\frac{1^{4 - 1}}{x - 5}$

Step 3.3

Subtract $1$ from $4$ .

$\frac{1^{3}}{x - 5}$

Step 3.4

One to any power is one.

$\frac{1}{x - 5}$