Simplify limit as x approaches 2 of (x^5-32)/(x^2+3x-10)

$lim_{x \to 2} \frac{x^{5} - 32}{x^{2} + 3 x - 10}$

Step 1

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.

$\frac{lim_{x \to 2} x^{5} - 32}{lim_{x \to 2} x^{2} + 3 x - 10}$

Evaluate the limit of the numerator.

Evaluate the limit.

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

$\frac{lim_{x \to 2} x^{5} - lim_{x \to 2} 32}{lim_{x \to 2} x^{2} + 3 x - 10}$

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

$\frac{{(lim_{x \to 2} x)}^{5} - lim_{x \to 2} 32}{lim_{x \to 2} x^{2} + 3 x - 10}$

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

$\frac{{(lim_{x \to 2} x)}^{5} - 1 \cdot 32}{lim_{x \to 2} x^{2} + 3 x - 10}$

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

$\frac{2^{5} - 1 \cdot 32}{lim_{x \to 2} x^{2} + 3 x - 10}$

Simplify the answer.

Simplify each term.

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

$\frac{32 - 1 \cdot 32}{lim_{x \to 2} x^{2} + 3 x - 10}$

Multiply $- 1$ by $32$ .

$\frac{32 - 32}{lim_{x \to 2} x^{2} + 3 x - 10}$

Subtract $32$ from $32$ .

$\frac{0}{lim_{x \to 2} x^{2} + 3 x - 10}$

Evaluate the limit of the denominator.

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

$\frac{0}{lim_{x \to 2} x^{2} + lim_{x \to 2} 3 x - lim_{x \to 2} 10}$

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

$\frac{0}{{(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 3 x - lim_{x \to 2} 10}$

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x - lim_{x \to 2} 10}$

Evaluate the limit of $10$ which is constant as $x$ approaches $2$ .

$\frac{0}{{(lim_{x \to 2} x)}^{2} + 3 lim_{x \to 2} x - 1 \cdot 10}$

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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

$\frac{0}{2^{2} + 3 lim_{x \to 2} x - 1 \cdot 10}$

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

$\frac{0}{2^{2} + 3 \cdot 2 - 1 \cdot 10}$

Simplify the answer.

Simplify each term.

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

$\frac{0}{4 + 3 \cdot 2 - 1 \cdot 10}$

Multiply $3$ by $2$ .

$\frac{0}{4 + 6 - 1 \cdot 10}$

Multiply $- 1$ by $10$ .

$\frac{0}{4 + 6 - 10}$

Add $4$ and $6$ .

$\frac{0}{10 - 10}$

Subtract $10$ from $10$ .

$\frac{0}{0}$

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.

$lim_{x \to 2} \frac{x^{5} - 32}{x^{2} + 3 x - 10} = lim_{x \to 2} \frac{\frac{d}{d x} [x^{5} - 32]}{\frac{d}{d x} [x^{2} + 3 x - 10]}$

Find the derivative of the numerator and denominator.

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{5} - 32]}{\frac{d}{d x} [x^{2} + 3 x - 10]}$

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

$lim_{x \to 2} \frac{\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 32]}{\frac{d}{d x} [x^{2} + 3 x - 10]}$

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

$lim_{x \to 2} \frac{5 x^{4} + \frac{d}{d x} [- 32]}{\frac{d}{d x} [x^{2} + 3 x - 10]}$

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

$lim_{x \to 2} \frac{5 x^{4} + 0}{\frac{d}{d x} [x^{2} + 3 x - 10]}$

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

$lim_{x \to 2} \frac{5 x^{4}}{\frac{d}{d x} [x^{2} + 3 x - 10]}$

By the Sum Rule, the derivative of $x^{2} + 3 x - 10$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 10]$ .

$lim_{x \to 2} \frac{5 x^{4}}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 10]}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 2} \frac{5 x^{4}}{2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 10]}$

Evaluate $\frac{d}{d x} [3 x]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$lim_{x \to 2} \frac{5 x^{4}}{2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 10]}$

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

$lim_{x \to 2} \frac{5 x^{4}}{2 x + 3 \cdot 1 + \frac{d}{d x} [- 10]}$

Multiply $3$ by $1$ .

$lim_{x \to 2} \frac{5 x^{4}}{2 x + 3 + \frac{d}{d x} [- 10]}$

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

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

Add $2 x + 3$ and $0$ .

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

Step 2

Evaluate the limit.

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

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

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

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

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

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

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

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

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Evaluate the limit of $3$ which is constant as $x$ approaches $2$ .

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

Step 3

Simplify the numerator.

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

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

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

$5 \frac{16}{2 lim_{x \to 2} x + 3}$

Step 4

Simplify the denominator.

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

$5 \frac{16}{2 \cdot 2 + 3}$

Multiply $2$ by $2$ .

$5 \frac{16}{4 + 3}$

Add $4$ and $3$ .

$5 (\frac{16}{7})$

Step 5

Multiply $5 (\frac{16}{7})$ .

Combine $5$ and $\frac{16}{7}$ .

$\frac{5 \cdot 16}{7}$

Multiply $5$ by $16$ .

$\frac{80}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{80}{7}$

Decimal Form:

$11.42857142 \dots$

Simplify limit as x approaches 2 of (x^5-32)/(x^2+3x-10)

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