Calculus Examples

$lim_{x \to 8} \frac{\sqrt{9 x^{2} + 16}}{2 + \sqrt{x^{3} + 1}}$

Step 1

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

$\frac{lim_{x \to 8} \sqrt{9 x^{2} + 16}}{lim_{x \to 8} 2 + \sqrt{x^{3} + 1}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 8} 9 x^{2} + 16}}{lim_{x \to 8} 2 + \sqrt{x^{3} + 1}}$

Step 3

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

$\frac{\sqrt{lim_{x \to 8} 9 x^{2} + lim_{x \to 8} 16}}{lim_{x \to 8} 2 + \sqrt{x^{3} + 1}}$

Step 4

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

$\frac{\sqrt{9 lim_{x \to 8} x^{2} + lim_{x \to 8} 16}}{lim_{x \to 8} 2 + \sqrt{x^{3} + 1}}$

Step 5

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

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 16}}{lim_{x \to 8} 2 + \sqrt{x^{3} + 1}}$

Step 6

Evaluate the limit of $16$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 16}}{lim_{x \to 8} 2 + \sqrt{x^{3} + 1}}$

Step 7

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

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 16}}{lim_{x \to 8} 2 + lim_{x \to 8} \sqrt{x^{3} + 1}}$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 16}}{2 + lim_{x \to 8} \sqrt{x^{3} + 1}}$

Step 9

Move the limit under the radical sign.

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 16}}{2 + \sqrt{lim_{x \to 8} x^{3} + 1}}$

Step 10

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

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 16}}{2 + \sqrt{lim_{x \to 8} x^{3} + lim_{x \to 8} 1}}$

Step 11

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 16}}{2 + \sqrt{{(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 1}}$

Step 12

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{\sqrt{9 {(lim_{x \to 8} x)}^{2} + 16}}{2 + \sqrt{{(lim_{x \to 8} x)}^{3} + 1}}$

Step 13

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

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Step 13.1

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

$\frac{\sqrt{9 \cdot 8^{2} + 16}}{2 + \sqrt{{(lim_{x \to 8} x)}^{3} + 1}}$

Step 13.2

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

$\frac{\sqrt{9 \cdot 8^{2} + 16}}{2 + \sqrt{8^{3} + 1}}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

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

$\frac{\sqrt{9 \cdot 64 + 16}}{2 + \sqrt{8^{3} + 1}}$

Step 14.1.2

Multiply $9$ by $64$ .

$\frac{\sqrt{576 + 16}}{2 + \sqrt{8^{3} + 1}}$

Step 14.1.3

Add $576$ and $16$ .

$\frac{\sqrt{592}}{2 + \sqrt{8^{3} + 1}}$

Step 14.1.4

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

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Step 14.1.4.1

Factor $16$ out of $592$ .

$\frac{\sqrt{16 (37)}}{2 + \sqrt{8^{3} + 1}}$

Step 14.1.4.2

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

$\frac{\sqrt{4^{2} \cdot 37}}{2 + \sqrt{8^{3} + 1}}$

Step 14.1.5

Pull terms out from under the radical.

$\frac{4 \sqrt{37}}{2 + \sqrt{8^{3} + 1}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Raise $8$ to the power of $3$ .

$\frac{4 \sqrt{37}}{2 + \sqrt{512 + 1}}$

Step 14.2.2

Add $512$ and $1$ .

$\frac{4 \sqrt{37}}{2 + \sqrt{513}}$

Step 14.2.3

Rewrite $513$ as $3^{2} \cdot 57$ .

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Step 14.2.3.1

Factor $9$ out of $513$ .

$\frac{4 \sqrt{37}}{2 + \sqrt{9 (57)}}$

Step 14.2.3.2

Rewrite $9$ as $3^{2}$ .

$\frac{4 \sqrt{37}}{2 + \sqrt{3^{2} \cdot 57}}$

Step 14.2.4

Pull terms out from under the radical.

$\frac{4 \sqrt{37}}{2 + 3 \sqrt{57}}$

Step 14.3

Multiply $\frac{4 \sqrt{37}}{2 + 3 \sqrt{57}}$ by $\frac{2 - 3 \sqrt{57}}{2 - 3 \sqrt{57}}$ .

$\frac{4 \sqrt{37}}{2 + 3 \sqrt{57}} \cdot \frac{2 - 3 \sqrt{57}}{2 - 3 \sqrt{57}}$

Step 14.4

Multiply $\frac{4 \sqrt{37}}{2 + 3 \sqrt{57}}$ by $\frac{2 - 3 \sqrt{57}}{2 - 3 \sqrt{57}}$ .

$\frac{4 \sqrt{37} (2 - 3 \sqrt{57})}{(2 + 3 \sqrt{57}) (2 - 3 \sqrt{57})}$

Step 14.5

Expand the denominator using the FOIL method.

$\frac{4 \sqrt{37} (2 - 3 \sqrt{57})}{4 - 6 \sqrt{57} + 6 \sqrt{57} - 9 {\sqrt{57}}^{2}}$

Step 14.6

Simplify.

$\frac{4 \sqrt{37} (2 - 3 \sqrt{57})}{- 509}$

Step 14.7

Group $2 - 3 \sqrt{57}$ and $\sqrt{37}$ together.

$\frac{4 ((2 - 3 \sqrt{57}) \sqrt{37})}{- 509}$

Step 14.8

Apply the distributive property.

$\frac{4 (2 \sqrt{37} - 3 \sqrt{57} \sqrt{37})}{- 509}$

Step 14.9

Multiply $- 3 \sqrt{57} \sqrt{37}$ .

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Step 14.9.1

Combine using the product rule for radicals.

$\frac{4 (2 \sqrt{37} - 3 \sqrt{37 \cdot 57})}{- 509}$

Step 14.9.2

Multiply $37$ by $57$ .

$\frac{4 (2 \sqrt{37} - 3 \sqrt{2109})}{- 509}$

Step 14.10

Move the negative in front of the fraction.

$- \frac{4 (2 \sqrt{37} - 3 \sqrt{2109})}{509}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 (2 \sqrt{37} - 3 \sqrt{2109})}{509}$

Decimal Form:

$0.98708074 \dots$