Calculus Examples

$lim_{x \to 8} \frac{4 x^{2}}{\sqrt{9 x^{4} - 1}}$

Step 1

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

$4 lim_{x \to 8} \frac{x^{2}}{\sqrt{9 x^{4} - 1}}$

Step 2

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

$4 \frac{lim_{x \to 8} x^{2}}{lim_{x \to 8} \sqrt{9 x^{4} - 1}}$

Step 3

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

$4 \frac{{(lim_{x \to 8} x)}^{2}}{lim_{x \to 8} \sqrt{9 x^{4} - 1}}$

Step 4

Move the limit under the radical sign.

$4 \frac{{(lim_{x \to 8} x)}^{2}}{\sqrt{lim_{x \to 8} 9 x^{4} - 1}}$

Step 5

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

$4 \frac{{(lim_{x \to 8} x)}^{2}}{\sqrt{lim_{x \to 8} 9 x^{4} - lim_{x \to 8} 1}}$

Step 6

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

$4 \frac{{(lim_{x \to 8} x)}^{2}}{\sqrt{9 lim_{x \to 8} x^{4} - lim_{x \to 8} 1}}$

Step 7

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$4 \frac{{(lim_{x \to 8} x)}^{2}}{\sqrt{9 {(lim_{x \to 8} x)}^{4} - lim_{x \to 8} 1}}$

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

$4 \frac{8^{2}}{\sqrt{9 \cdot 8^{4} - 1 \cdot 1}}$

Step 10

Simplify the answer.

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Step 10.1

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

$4 \frac{64}{\sqrt{9 \cdot 8^{4} - 1 \cdot 1}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Raise $8$ to the power of $4$ .

$4 \frac{64}{\sqrt{9 \cdot 4096 - 1 \cdot 1}}$

Step 10.2.2

Multiply $9$ by $4096$ .

$4 \frac{64}{\sqrt{36864 - 1 \cdot 1}}$

Step 10.2.3

Multiply $- 1$ by $1$ .

$4 \frac{64}{\sqrt{36864 - 1}}$

Step 10.2.4

Subtract $1$ from $36864$ .

$4 \frac{64}{\sqrt{36863}}$

Step 10.3

Multiply $\frac{64}{\sqrt{36863}}$ by $\frac{\sqrt{36863}}{\sqrt{36863}}$ .

$4 (\frac{64}{\sqrt{36863}} \cdot \frac{\sqrt{36863}}{\sqrt{36863}})$

Step 10.4

Combine and simplify the denominator.

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Step 10.4.1

Multiply $\frac{64}{\sqrt{36863}}$ by $\frac{\sqrt{36863}}{\sqrt{36863}}$ .

$4 \frac{64 \sqrt{36863}}{\sqrt{36863} \sqrt{36863}}$

Step 10.4.2

Raise $\sqrt{36863}$ to the power of $1$ .

$4 \frac{64 \sqrt{36863}}{{\sqrt{36863}}^{1} \sqrt{36863}}$

Step 10.4.3

Raise $\sqrt{36863}$ to the power of $1$ .

$4 \frac{64 \sqrt{36863}}{{\sqrt{36863}}^{1} {\sqrt{36863}}^{1}}$

Step 10.4.4

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

$4 \frac{64 \sqrt{36863}}{{\sqrt{36863}}^{1 + 1}}$

Step 10.4.5

Add $1$ and $1$ .

$4 \frac{64 \sqrt{36863}}{{\sqrt{36863}}^{2}}$

Step 10.4.6

Rewrite ${\sqrt{36863}}^{2}$ as $36863$ .

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Step 10.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{36863}$ as $36863^{\frac{1}{2}}$ .

$4 \frac{64 \sqrt{36863}}{{(36863^{\frac{1}{2}})}^{2}}$

Step 10.4.6.2

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

$4 \frac{64 \sqrt{36863}}{36863^{\frac{1}{2} \cdot 2}}$

Step 10.4.6.3

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

$4 \frac{64 \sqrt{36863}}{36863^{\frac{2}{2}}}$

Step 10.4.6.4

Cancel the common factor of $2$ .

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Step 10.4.6.4.1

Cancel the common factor.

$4 \frac{64 \sqrt{36863}}{36863^{\frac{2}{2}}}$

Step 10.4.6.4.2

Rewrite the expression.

$4 \frac{64 \sqrt{36863}}{36863^{1}}$

Step 10.4.6.5

Evaluate the exponent.

$4 \frac{64 \sqrt{36863}}{36863}$

Step 10.5

Multiply $4 \frac{64 \sqrt{36863}}{36863}$ .

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Step 10.5.1

Combine $4$ and $\frac{64 \sqrt{36863}}{36863}$ .

$\frac{4 (64 \sqrt{36863})}{36863}$

Step 10.5.2

Multiply $64$ by $4$ .

$\frac{256 \sqrt{36863}}{36863}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{256 \sqrt{36863}}{36863}$

Decimal Form:

$1.33335141 \dots$