Calculus Examples

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

Step 1

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

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

Step 2

Move the limit under the radical sign.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the denominator.

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Step 8.1.1

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

$\frac{8}{\sqrt{4 \cdot 64 + 1}}$

Step 8.1.2

Multiply $4$ by $64$ .

$\frac{8}{\sqrt{256 + 1}}$

Step 8.1.3

Add $256$ and $1$ .

$\frac{8}{\sqrt{257}}$

Step 8.2

Multiply $\frac{8}{\sqrt{257}}$ by $\frac{\sqrt{257}}{\sqrt{257}}$ .

$\frac{8}{\sqrt{257}} \cdot \frac{\sqrt{257}}{\sqrt{257}}$

Step 8.3

Combine and simplify the denominator.

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Step 8.3.1

Multiply $\frac{8}{\sqrt{257}}$ by $\frac{\sqrt{257}}{\sqrt{257}}$ .

$\frac{8 \sqrt{257}}{\sqrt{257} \sqrt{257}}$

Step 8.3.2

Raise $\sqrt{257}$ to the power of $1$ .

$\frac{8 \sqrt{257}}{{\sqrt{257}}^{1} \sqrt{257}}$

Step 8.3.3

Raise $\sqrt{257}$ to the power of $1$ .

$\frac{8 \sqrt{257}}{{\sqrt{257}}^{1} {\sqrt{257}}^{1}}$

Step 8.3.4

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

$\frac{8 \sqrt{257}}{{\sqrt{257}}^{1 + 1}}$

Step 8.3.5

Add $1$ and $1$ .

$\frac{8 \sqrt{257}}{{\sqrt{257}}^{2}}$

Step 8.3.6

Rewrite ${\sqrt{257}}^{2}$ as $257$ .

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Step 8.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{257}$ as $257^{\frac{1}{2}}$ .

$\frac{8 \sqrt{257}}{{(257^{\frac{1}{2}})}^{2}}$

Step 8.3.6.2

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

$\frac{8 \sqrt{257}}{257^{\frac{1}{2} \cdot 2}}$

Step 8.3.6.3

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

$\frac{8 \sqrt{257}}{257^{\frac{2}{2}}}$

Step 8.3.6.4

Cancel the common factor of $2$ .

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Step 8.3.6.4.1

Cancel the common factor.

$\frac{8 \sqrt{257}}{257^{\frac{2}{2}}}$

Step 8.3.6.4.2

Rewrite the expression.

$\frac{8 \sqrt{257}}{257^{1}}$

Step 8.3.6.5

Evaluate the exponent.

$\frac{8 \sqrt{257}}{257}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{8 \sqrt{257}}{257}$

Decimal Form:

$0.49902628 \dots$