Calculus Examples

$lim_{x \to - 8} x + \sqrt{x^{2} + 6 x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$lim_{x \to - 8} x + lim_{x \to - 8} \sqrt{x^{2} + 6 x}$

Step 2

Move the limit under the radical sign.

$lim_{x \to - 8} x + \sqrt{lim_{x \to - 8} x^{2} + 6 x}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$lim_{x \to - 8} x + \sqrt{lim_{x \to - 8} x^{2} + lim_{x \to - 8} 6 x}$

Step 4

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

$lim_{x \to - 8} x + \sqrt{{(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} 6 x}$

Step 5

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to - 8} x + \sqrt{{(lim_{x \to - 8} x)}^{2} + 6 lim_{x \to - 8} x}$

Step 6

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- 8 + \sqrt{{(lim_{x \to - 8} x)}^{2} + 6 lim_{x \to - 8} x}$

Step 6.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- 8 + \sqrt{{(- 8)}^{2} + 6 lim_{x \to - 8} x}$

Step 6.3

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$- 8 + \sqrt{{(- 8)}^{2} + 6 \cdot - 8}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Raise $- 8$ to the power of $2$ .

$- 8 + \sqrt{64 + 6 \cdot - 8}$

Step 7.1.2

Multiply $6$ by $- 8$ .

$- 8 + \sqrt{64 - 48}$

Step 7.1.3

Subtract $48$ from $64$ .

$- 8 + \sqrt{16}$

Step 7.1.4

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

$- 8 + \sqrt{4^{2}}$

Step 7.1.5

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

$- 8 + 4$

Step 7.2

Add $- 8$ and $4$ .

$- 4$