Calculus Examples

$lim_{x \to - 2} \sqrt{x^{4} + 3 x + 6}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to - 2} x^{4} + 3 x + 6}$

Step 2

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

$\sqrt{lim_{x \to - 2} x^{4} + lim_{x \to - 2} 3 x + lim_{x \to - 2} 6}$

Step 3

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

$\sqrt{{(lim_{x \to - 2} x)}^{4} + lim_{x \to - 2} 3 x + lim_{x \to - 2} 6}$

Step 4

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

$\sqrt{{(lim_{x \to - 2} x)}^{4} + 3 lim_{x \to - 2} x + lim_{x \to - 2} 6}$

Step 5

Evaluate the limit of $6$ which is constant as $x$ approaches $- 2$ .

$\sqrt{{(lim_{x \to - 2} x)}^{4} + 3 lim_{x \to - 2} x + 6}$

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 $x$ .

$\sqrt{{(- 2)}^{4} + 3 lim_{x \to - 2} x + 6}$

Step 6.2

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

$\sqrt{{(- 2)}^{4} + 3 \cdot - 2 + 6}$

Step 7

Simplify the answer.

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Step 7.1

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

$\sqrt{16 + 3 \cdot - 2 + 6}$

Step 7.2

Multiply $3$ by $- 2$ .

$\sqrt{16 - 6 + 6}$

Step 7.3

Subtract $6$ from $16$ .

$\sqrt{10 + 6}$

Step 7.4

Add $10$ and $6$ .

$\sqrt{16}$

Step 7.5

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

Step 7.6

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

$4$