Calculus Examples

$lim_{x \to - 8} 2 x + \sqrt{4 x^{2} + 2 x - 4}$

Step 1

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

$lim_{x \to - 8} 2 x + lim_{x \to - 8} \sqrt{4 x^{2} + 2 x - 4}$

Step 2

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

$2 lim_{x \to - 8} x + lim_{x \to - 8} \sqrt{4 x^{2} + 2 x - 4}$

Step 3

Move the limit under the radical sign.

$2 lim_{x \to - 8} x + \sqrt{lim_{x \to - 8} 4 x^{2} + 2 x - 4}$

Step 4

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

$2 lim_{x \to - 8} x + \sqrt{lim_{x \to - 8} 4 x^{2} + lim_{x \to - 8} 2 x - lim_{x \to - 8} 4}$

Step 5

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

$2 lim_{x \to - 8} x + \sqrt{4 lim_{x \to - 8} x^{2} + lim_{x \to - 8} 2 x - lim_{x \to - 8} 4}$

Step 6

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

$2 lim_{x \to - 8} x + \sqrt{4 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} 2 x - lim_{x \to - 8} 4}$

Step 7

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

$2 lim_{x \to - 8} x + \sqrt{4 {(lim_{x \to - 8} x)}^{2} + 2 lim_{x \to - 8} x - lim_{x \to - 8} 4}$

Step 8

Evaluate the limit of $4$ which is constant as $x$ approaches $- 8$ .

$2 lim_{x \to - 8} x + \sqrt{4 {(lim_{x \to - 8} x)}^{2} + 2 lim_{x \to - 8} x - 1 \cdot 4}$

Step 9

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

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Step 9.1

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

$2 \cdot - 8 + \sqrt{4 {(lim_{x \to - 8} x)}^{2} + 2 lim_{x \to - 8} x - 1 \cdot 4}$

Step 9.2

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

$2 \cdot - 8 + \sqrt{4 {(- 8)}^{2} + 2 lim_{x \to - 8} x - 1 \cdot 4}$

Step 9.3

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

$2 \cdot - 8 + \sqrt{4 {(- 8)}^{2} + 2 \cdot - 8 - 1 \cdot 4}$

Step 10

Simplify each term.

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Step 10.1

Multiply $2$ by $- 8$ .

$- 16 + \sqrt{4 {(- 8)}^{2} + 2 \cdot - 8 - 1 \cdot 4}$

Step 10.2

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

$- 16 + \sqrt{4 \cdot 64 + 2 \cdot - 8 - 1 \cdot 4}$

Step 10.3

Multiply $4$ by $64$ .

$- 16 + \sqrt{256 + 2 \cdot - 8 - 1 \cdot 4}$

Step 10.4

Multiply $2$ by $- 8$ .

$- 16 + \sqrt{256 - 16 - 1 \cdot 4}$

Step 10.5

Multiply $- 1$ by $4$ .

$- 16 + \sqrt{256 - 16 - 4}$

Step 10.6

Subtract $16$ from $256$ .

$- 16 + \sqrt{240 - 4}$

Step 10.7

Subtract $4$ from $240$ .

$- 16 + \sqrt{236}$

Step 10.8

Rewrite $236$ as $2^{2} \cdot 59$ .

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Step 10.8.1

Factor $4$ out of $236$ .

$- 16 + \sqrt{4 (59)}$

Step 10.8.2

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

$- 16 + \sqrt{2^{2} \cdot 59}$

Step 10.9

Pull terms out from under the radical.

$- 16 + 2 \sqrt{59}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- 16 + 2 \sqrt{59}$

Decimal Form:

$- 0.63770850 \dots$