Calculus Examples

$lim_{x \to - 2} \sqrt{7 x^{4} + x^{2}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to - 2} 7 x^{4} + x^{2}}$

Step 2

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

$\sqrt{lim_{x \to - 2} 7 x^{4} + lim_{x \to - 2} x^{2}}$

Step 3

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{7 lim_{x \to - 2} x^{4} + lim_{x \to - 2} x^{2}}$

Step 4

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

$\sqrt{7 {(lim_{x \to - 2} x)}^{4} + lim_{x \to - 2} x^{2}}$

Step 5

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

$\sqrt{7 {(lim_{x \to - 2} x)}^{4} + {(lim_{x \to - 2} x)}^{2}}$

Step 6

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

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Step 6.1

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

$\sqrt{7 {(- 2)}^{4} + {(lim_{x \to - 2} x)}^{2}}$

Step 6.2

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

$\sqrt{7 {(- 2)}^{4} + {(- 2)}^{2}}$

Step 7

Simplify the answer.

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Step 7.1

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

$\sqrt{7 \cdot 16 + {(- 2)}^{2}}$

Step 7.2

Multiply $7$ by $16$ .

$\sqrt{112 + {(- 2)}^{2}}$

Step 7.3

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

$\sqrt{112 + 4}$

Step 7.4

Add $112$ and $4$ .

$\sqrt{116}$

Step 7.5

Rewrite $116$ as $2^{2} \cdot 29$ .

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Step 7.5.1

Factor $4$ out of $116$ .

$\sqrt{4 (29)}$

Step 7.5.2

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 29}$

Step 7.6

Pull terms out from under the radical.

$2 \sqrt{29}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{29}$

Decimal Form:

$10.77032961 \dots$