Calculus Examples

$lim_{x \to 8} \frac{4 x^{2} + 9}{\sqrt{x^{7} + 6 x^{2} - 5}}$

Step 1

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

$\frac{lim_{x \to 8} 4 x^{2} + 9}{lim_{x \to 8} \sqrt{x^{7} + 6 x^{2} - 5}}$

Step 2

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

$\frac{lim_{x \to 8} 4 x^{2} + lim_{x \to 8} 9}{lim_{x \to 8} \sqrt{x^{7} + 6 x^{2} - 5}}$

Step 3

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

$\frac{4 lim_{x \to 8} x^{2} + lim_{x \to 8} 9}{lim_{x \to 8} \sqrt{x^{7} + 6 x^{2} - 5}}$

Step 4

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

$\frac{4 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 9}{lim_{x \to 8} \sqrt{x^{7} + 6 x^{2} - 5}}$

Step 5

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$\frac{4 {(lim_{x \to 8} x)}^{2} + 9}{lim_{x \to 8} \sqrt{x^{7} + 6 x^{2} - 5}}$

Step 6

Move the limit under the radical sign.

$\frac{4 {(lim_{x \to 8} x)}^{2} + 9}{\sqrt{lim_{x \to 8} x^{7} + 6 x^{2} - 5}}$

Step 7

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

$\frac{4 {(lim_{x \to 8} x)}^{2} + 9}{\sqrt{lim_{x \to 8} x^{7} + lim_{x \to 8} 6 x^{2} - lim_{x \to 8} 5}}$

Step 8

Move the exponent $7$ from $x^{7}$ outside the limit using the Limits Power Rule.

$\frac{4 {(lim_{x \to 8} x)}^{2} + 9}{\sqrt{{(lim_{x \to 8} x)}^{7} + lim_{x \to 8} 6 x^{2} - lim_{x \to 8} 5}}$

Step 9

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 {(lim_{x \to 8} x)}^{2} + 9}{\sqrt{{(lim_{x \to 8} x)}^{7} + 6 lim_{x \to 8} x^{2} - lim_{x \to 8} 5}}$

Step 10

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

$\frac{4 {(lim_{x \to 8} x)}^{2} + 9}{\sqrt{{(lim_{x \to 8} x)}^{7} + 6 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 5}}$

Step 11

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{4 {(lim_{x \to 8} x)}^{2} + 9}{\sqrt{{(lim_{x \to 8} x)}^{7} + 6 {(lim_{x \to 8} x)}^{2} - 1 \cdot 5}}$

Step 12

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

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Step 12.1

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

$\frac{4 \cdot 8^{2} + 9}{\sqrt{{(lim_{x \to 8} x)}^{7} + 6 {(lim_{x \to 8} x)}^{2} - 1 \cdot 5}}$

Step 12.2

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

$\frac{4 \cdot 8^{2} + 9}{\sqrt{8^{7} + 6 {(lim_{x \to 8} x)}^{2} - 1 \cdot 5}}$

Step 12.3

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

$\frac{4 \cdot 8^{2} + 9}{\sqrt{8^{7} + 6 \cdot 8^{2} - 1 \cdot 5}}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{4 \cdot 64 + 9}{\sqrt{8^{7} + 6 \cdot 8^{2} - 1 \cdot 5}}$

Step 13.1.2

Multiply $4$ by $64$ .

$\frac{256 + 9}{\sqrt{8^{7} + 6 \cdot 8^{2} - 1 \cdot 5}}$

Step 13.1.3

Add $256$ and $9$ .

$\frac{265}{\sqrt{8^{7} + 6 \cdot 8^{2} - 1 \cdot 5}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Raise $8$ to the power of $7$ .

$\frac{265}{\sqrt{2097152 + 6 \cdot 8^{2} - 1 \cdot 5}}$

Step 13.2.2

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

$\frac{265}{\sqrt{2097152 + 6 \cdot 64 - 1 \cdot 5}}$

Step 13.2.3

Multiply $6$ by $64$ .

$\frac{265}{\sqrt{2097152 + 384 - 1 \cdot 5}}$

Step 13.2.4

Multiply $- 1$ by $5$ .

$\frac{265}{\sqrt{2097152 + 384 - 5}}$

Step 13.2.5

Add $2097152$ and $384$ .

$\frac{265}{\sqrt{2097536 - 5}}$

Step 13.2.6

Subtract $5$ from $2097536$ .

$\frac{265}{\sqrt{2097531}}$

Step 13.2.7

Rewrite $2097531$ as $3^{2} \cdot 233059$ .

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Step 13.2.7.1

Factor $9$ out of $2097531$ .

$\frac{265}{\sqrt{9 (233059)}}$

Step 13.2.7.2

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

$\frac{265}{\sqrt{3^{2} \cdot 233059}}$

Step 13.2.8

Pull terms out from under the radical.

$\frac{265}{3 \sqrt{233059}}$

Step 13.3

Multiply $\frac{265}{3 \sqrt{233059}}$ by $\frac{\sqrt{233059}}{\sqrt{233059}}$ .

$\frac{265}{3 \sqrt{233059}} \cdot \frac{\sqrt{233059}}{\sqrt{233059}}$

Step 13.4

Combine and simplify the denominator.

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Step 13.4.1

Multiply $\frac{265}{3 \sqrt{233059}}$ by $\frac{\sqrt{233059}}{\sqrt{233059}}$ .

$\frac{265 \sqrt{233059}}{3 \sqrt{233059} \sqrt{233059}}$

Step 13.4.2

Move $\sqrt{233059}$ .

$\frac{265 \sqrt{233059}}{3 (\sqrt{233059} \sqrt{233059})}$

Step 13.4.3

Raise $\sqrt{233059}$ to the power of $1$ .

$\frac{265 \sqrt{233059}}{3 ({\sqrt{233059}}^{1} \sqrt{233059})}$

Step 13.4.4

Raise $\sqrt{233059}$ to the power of $1$ .

$\frac{265 \sqrt{233059}}{3 ({\sqrt{233059}}^{1} {\sqrt{233059}}^{1})}$

Step 13.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{265 \sqrt{233059}}{3 {\sqrt{233059}}^{1 + 1}}$

Step 13.4.6

Add $1$ and $1$ .

$\frac{265 \sqrt{233059}}{3 {\sqrt{233059}}^{2}}$

Step 13.4.7

Rewrite ${\sqrt{233059}}^{2}$ as $233059$ .

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Step 13.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{233059}$ as $233059^{\frac{1}{2}}$ .

$\frac{265 \sqrt{233059}}{3 {(233059^{\frac{1}{2}})}^{2}}$

Step 13.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{265 \sqrt{233059}}{3 \cdot 233059^{\frac{1}{2} \cdot 2}}$

Step 13.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{265 \sqrt{233059}}{3 \cdot 233059^{\frac{2}{2}}}$

Step 13.4.7.4

Cancel the common factor of $2$ .

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Step 13.4.7.4.1

Cancel the common factor.

$\frac{265 \sqrt{233059}}{3 \cdot 233059^{\frac{2}{2}}}$

Step 13.4.7.4.2

Rewrite the expression.

$\frac{265 \sqrt{233059}}{3 \cdot 233059^{1}}$

Step 13.4.7.5

Evaluate the exponent.

$\frac{265 \sqrt{233059}}{3 \cdot 233059}$

Step 13.5

Multiply $3$ by $233059$ .

$\frac{265 \sqrt{233059}}{699177}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{265 \sqrt{233059}}{699177}$

Decimal Form:

$0.18297496 \dots$