Calculus Examples

$lim_{x \to - 3} x^{2} - 16 + \sqrt[3]{x^{2} - 36}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$lim_{x \to - 3} x^{2} - lim_{x \to - 3} 16 + lim_{x \to - 3} \sqrt[3]{x^{2} - 36}$

Step 2

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

${(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} 16 + lim_{x \to - 3} \sqrt[3]{x^{2} - 36}$

Step 3

Evaluate the limit of $16$ which is constant as $x$ approaches $- 3$ .

${(lim_{x \to - 3} x)}^{2} - 1 \cdot 16 + lim_{x \to - 3} \sqrt[3]{x^{2} - 36}$

Step 4

Move the limit under the radical sign.

${(lim_{x \to - 3} x)}^{2} - 1 \cdot 16 + \sqrt[3]{lim_{x \to - 3} x^{2} - 36}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

${(lim_{x \to - 3} x)}^{2} - 1 \cdot 16 + \sqrt[3]{lim_{x \to - 3} x^{2} - lim_{x \to - 3} 36}$

Step 6

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

${(lim_{x \to - 3} x)}^{2} - 1 \cdot 16 + \sqrt[3]{{(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} 36}$

Step 7

Evaluate the limit of $36$ which is constant as $x$ approaches $- 3$ .

${(lim_{x \to - 3} x)}^{2} - 1 \cdot 16 + \sqrt[3]{{(lim_{x \to - 3} x)}^{2} - 1 \cdot 36}$

Step 8

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

${(- 3)}^{2} - 1 \cdot 16 + \sqrt[3]{{(lim_{x \to - 3} x)}^{2} - 1 \cdot 36}$

Step 8.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

${(- 3)}^{2} - 1 \cdot 16 + \sqrt[3]{{(- 3)}^{2} - 1 \cdot 36}$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$9 - 1 \cdot 16 + \sqrt[3]{{(- 3)}^{2} - 1 \cdot 36}$

Step 9.1.2

Multiply $- 1$ by $16$ .

$9 - 16 + \sqrt[3]{{(- 3)}^{2} - 1 \cdot 36}$

Step 9.1.3

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

$9 - 16 + \sqrt[3]{9 - 1 \cdot 36}$

Step 9.1.4

Multiply $- 1$ by $36$ .

$9 - 16 + \sqrt[3]{9 - 36}$

Step 9.1.5

Subtract $36$ from $9$ .

$9 - 16 + \sqrt[3]{- 27}$

Step 9.1.6

Rewrite $- 27$ as ${(- 3)}^{3}$ .

$9 - 16 + \sqrt[3]{{(- 3)}^{3}}$

Step 9.1.7

Pull terms out from under the radical, assuming real numbers.

$9 - 16 - 3$

Step 9.2

Subtract $16$ from $9$ .

$- 7 - 3$

Step 9.3

Subtract $3$ from $- 7$ .

$- 10$