Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 9

Simplify each term.

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Step 9.1

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

$\sqrt{9 \cdot 64 + 2 \cdot 8} - 3 \cdot 8$

Step 9.2

Multiply $9$ by $64$ .

$\sqrt{576 + 2 \cdot 8} - 3 \cdot 8$

Step 9.3

Multiply $2$ by $8$ .

$\sqrt{576 + 16} - 3 \cdot 8$

Step 9.4

Add $576$ and $16$ .

$\sqrt{592} - 3 \cdot 8$

Step 9.5

Rewrite $592$ as $4^{2} \cdot 37$ .

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Step 9.5.1

Factor $16$ out of $592$ .

$\sqrt{16 (37)} - 3 \cdot 8$

Step 9.5.2

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

$\sqrt{4^{2} \cdot 37} - 3 \cdot 8$

Step 9.6

Pull terms out from under the radical.

$4 \sqrt{37} - 3 \cdot 8$

Step 9.7

Multiply $- 3$ by $8$ .

$4 \sqrt{37} - 24$

Step 10

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{37} - 24$

Decimal Form:

$0.33105012 \dots$