Simplify limit as x approaches 2 of (x^5-32)/(x^3-8)

$lim_{x \to 2} \frac{x^{5} - 32}{x^{3} - 8}$

Apply L\'Hospital\'s rule.

Evaluate the limit of the numerator and the limit of the denominator.

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 2} x^{5} - 32}{lim_{x \to 2} x^{3} - 8}$

Evaluate the limit of the numerator.

Evaluate the limit.

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

$\frac{lim_{x \to 2} x^{5} - lim_{x \to 2} 32}{lim_{x \to 2} x^{3} - 8}$

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 2} x)}^{5} - lim_{x \to 2} 32}{lim_{x \to 2} x^{3} - 8}$

Evaluate the limit of $32$ which is constant as $x$ approaches $2$ .

$\frac{{(lim_{x \to 2} x)}^{5} - 1 \cdot 32}{lim_{x \to 2} x^{3} - 8}$

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

$\frac{2^{5} - 1 \cdot 32}{lim_{x \to 2} x^{3} - 8}$

Simplify the answer.

Simplify each term.

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

$\frac{32 - 1 \cdot 32}{lim_{x \to 2} x^{3} - 8}$

Multiply $- 1$ by $32$ .

$\frac{32 - 32}{lim_{x \to 2} x^{3} - 8}$

Subtract $32$ from $32$ .

$\frac{0}{lim_{x \to 2} x^{3} - 8}$

Evaluate the limit of the denominator.

Evaluate the limit.

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

$\frac{0}{lim_{x \to 2} x^{3} - lim_{x \to 2} 8}$

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 2} x)}^{3} - lim_{x \to 2} 8}$

Evaluate the limit of $8$ which is constant as $x$ approaches $2$ .

$\frac{0}{{(lim_{x \to 2} x)}^{3} - 1 \cdot 8}$

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

$\frac{0}{2^{3} - 1 \cdot 8}$

Simplify the answer.

Simplify each term.

Raise $2$ to the power of $3$ .

$\frac{0}{8 - 1 \cdot 8}$

Multiply $- 1$ by $8$ .

$\frac{0}{8 - 8}$

Subtract $8$ from $8$ .

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Since $\frac{0}{0}$ is of indeterminate form, apply L\'Hospital\'s Rule. L\'Hospital\'s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 2} \frac{x^{5} - 32}{x^{3} - 8} = lim_{x \to 2} \frac{\frac{d}{d x} [x^{5} - 32]}{\frac{d}{d x} [x^{3} - 8]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{5} - 32]}{\frac{d}{d x} [x^{3} - 8]}$

By the Sum Rule, the derivative of $x^{5} - 32$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 32]$ .

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 32]}{\frac{d}{d x} [x^{3} - 8]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$lim_{x \to 2} \frac{5 x^{4} + \frac{d}{d x} [- 32]}{\frac{d}{d x} [x^{3} - 8]}$

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{5 x^{4} + 0}{\frac{d}{d x} [x^{3} - 8]}$

Add $5 x^{4}$ and $0$ .

$lim_{x \to 2} \frac{5 x^{4}}{\frac{d}{d x} [x^{3} - 8]}$

By the Sum Rule, the derivative of $x^{3} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]$ .

$lim_{x \to 2} \frac{5 x^{4}}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to 2} \frac{5 x^{4}}{3 x^{2} + \frac{d}{d x} [- 8]}$

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{5 x^{4}}{3 x^{2} + 0}$

Add $3 x^{2}$ and $0$ .

$lim_{x \to 2} \frac{5 x^{4}}{3 x^{2}}$

Cancel the common factor of $x^{4}$ and $x^{2}$ .

Factor $x^{2}$ out of $5 x^{4}$ .

$lim_{x \to 2} \frac{x^{2} (5 x^{2})}{3 x^{2}}$

Cancel the common factors.

Factor $x^{2}$ out of $3 x^{2}$ .

$lim_{x \to 2} \frac{x^{2} (5 x^{2})}{x^{2} \cdot 3}$

Cancel the common factor.

$lim_{x \to 2} \frac{x^{2} (5 x^{2})}{x^{2} \cdot 3}$

Rewrite the expression.

$lim_{x \to 2} \frac{5 x^{2}}{3}$

Evaluate the limit.

Move the term $\frac{5}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{5}{3} lim_{x \to 2} x^{2}$

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

$\frac{5}{3} {(lim_{x \to 2} x)}^{2}$

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

$\frac{5}{3} \cdot 2^{2}$

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

$\frac{5}{3} \cdot 4$

Multiply $\frac{5}{3} \cdot 4$ .

Combine $\frac{5}{3}$ and $4$ .

$\frac{5 \cdot 4}{3}$

Multiply $5$ by $4$ .

$\frac{20}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{20}{3}$

Decimal Form:

$6.\overline{6}$

Simplify limit as x approaches 2 of (x^5-32)/(x^3-8)

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