Calculus Examples

$lim_{x \to - 2} \sqrt[3]{4 x^{2} - 25}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to - 2} 4 x^{2} - 25}$

Step 1.2

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

$\sqrt[3]{lim_{x \to - 2} 4 x^{2} - lim_{x \to - 2} 25}$

Step 1.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\sqrt[3]{4 lim_{x \to - 2} x^{2} - lim_{x \to - 2} 25}$

Step 1.4

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

$\sqrt[3]{4 {(lim_{x \to - 2} x)}^{2} - lim_{x \to - 2} 25}$

Step 1.5

Evaluate the limit of $25$ which is constant as $x$ approaches $- 2$ .

$\sqrt[3]{4 {(lim_{x \to - 2} x)}^{2} - 1 \cdot 25}$

Step 2

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

$\sqrt[3]{4 {(- 2)}^{2} - 1 \cdot 25}$

Step 3

Simplify the answer.

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Step 3.1

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

$\sqrt[3]{4 \cdot 4 - 1 \cdot 25}$

Step 3.2

Multiply $4$ by $4$ .

$\sqrt[3]{16 - 1 \cdot 25}$

Step 3.3

Multiply $- 1$ by $25$ .

$\sqrt[3]{16 - 25}$

Step 3.4

Subtract $25$ from $16$ .

$\sqrt[3]{- 9}$

Step 3.5

Rewrite $- 9$ as ${(- 1)}^{3} \cdot 9$ .

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Step 3.5.1

Rewrite $- 9$ as $- 1 (9)$ .

$\sqrt[3]{- 1 (9)}$

Step 3.5.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$\sqrt[3]{{(- 1)}^{3} \cdot 9}$

Step 3.6

Pull terms out from under the radical.

$- 1 \sqrt[3]{9}$

Step 3.7

Rewrite $- 1 \sqrt[3]{9}$ as $- \sqrt[3]{9}$ .

$- \sqrt[3]{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \sqrt[3]{9}$

Decimal Form:

$- 2.08008382 \dots$