Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

Simplify the answer.

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Step 7.1

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

$\sqrt{9 \cdot 64 - 8 - 3 \cdot 8}$

Step 7.2

Multiply $9$ by $64$ .

$\sqrt{576 - 8 - 3 \cdot 8}$

Step 7.3

Multiply $- 3$ by $8$ .

$\sqrt{576 - 8 - 24}$

Step 7.4

Subtract $8$ from $576$ .

$\sqrt{568 - 24}$

Step 7.5

Subtract $24$ from $568$ .

$\sqrt{544}$

Step 7.6

Rewrite $544$ as $4^{2} \cdot 34$ .

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Step 7.6.1

Factor $16$ out of $544$ .

$\sqrt{16 (34)}$

Step 7.6.2

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

$\sqrt{4^{2} \cdot 34}$

Step 7.7

Pull terms out from under the radical.

$4 \sqrt{34}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{34}$

Decimal Form:

$23.32380757 \dots$