Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

$\sqrt[3]{2 {(- 2)}^{2} - 2 + 2}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

$\sqrt[3]{2 \cdot 4 - 2 + 2}$

Step 7.2

Multiply $2$ by $4$ .

$\sqrt[3]{8 - 2 + 2}$

Step 7.3

Subtract $2$ from $8$ .

$\sqrt[3]{6 + 2}$

Step 7.4

Add $6$ and $2$ .

$\sqrt[3]{8}$

Step 7.5

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{2^{3}}$

Step 7.6

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

$2$