Calculus Examples

$lim_{x \to - 8} 3 x + \sqrt{9 x^{2} - x}$

Step 1

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

$lim_{x \to - 8} 3 x + lim_{x \to - 8} \sqrt{9 x^{2} - x}$

Step 2

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

$3 lim_{x \to - 8} x + lim_{x \to - 8} \sqrt{9 x^{2} - x}$

Step 3

Move the limit under the radical sign.

$3 lim_{x \to - 8} x + \sqrt{lim_{x \to - 8} 9 x^{2} - x}$

Step 4

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

$3 lim_{x \to - 8} x + \sqrt{lim_{x \to - 8} 9 x^{2} - lim_{x \to - 8} x}$

Step 5

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to - 8} x + \sqrt{9 lim_{x \to - 8} x^{2} - lim_{x \to - 8} x}$

Step 6

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

$3 lim_{x \to - 8} x + \sqrt{9 {(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} x}$

Step 7

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

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Step 7.1

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

$3 \cdot - 8 + \sqrt{9 {(lim_{x \to - 8} x)}^{2} - lim_{x \to - 8} x}$

Step 7.2

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

$3 \cdot - 8 + \sqrt{9 {(- 8)}^{2} - lim_{x \to - 8} x}$

Step 7.3

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

$3 \cdot - 8 + \sqrt{9 {(- 8)}^{2} + 8}$

Step 8

Simplify each term.

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Step 8.1

Multiply $3$ by $- 8$ .

$- 24 + \sqrt{9 {(- 8)}^{2} + 8}$

Step 8.2

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

$- 24 + \sqrt{9 \cdot 64 + 8}$

Step 8.3

Multiply $9$ by $64$ .

$- 24 + \sqrt{576 + 8}$

Step 8.4

Add $576$ and $8$ .

$- 24 + \sqrt{584}$

Step 8.5

Rewrite $584$ as $2^{2} \cdot 146$ .

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Step 8.5.1

Factor $4$ out of $584$ .

$- 24 + \sqrt{4 (146)}$

Step 8.5.2

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

$- 24 + \sqrt{2^{2} \cdot 146}$

Step 8.6

Pull terms out from under the radical.

$- 24 + 2 \sqrt{146}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 24 + 2 \sqrt{146}$

Decimal Form:

$0.16609194 \dots$