Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

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

Step 9

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

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

Step 10

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

Tap for more steps...

Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

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

$\sqrt{\frac{2 \cdot 64 - 1 \cdot 1}{8 + 8 \cdot 8^{2}}}$

Step 11.2

Multiply $2$ by $64$ .

$\sqrt{\frac{128 - 1 \cdot 1}{8 + 8 \cdot 8^{2}}}$

Step 11.3

Multiply $- 1$ by $1$ .

$\sqrt{\frac{128 - 1}{8 + 8 \cdot 8^{2}}}$

Step 11.4

Subtract $1$ from $128$ .

$\sqrt{\frac{127}{8 + 8 \cdot 8^{2}}}$

Step 11.5

Multiply $8$ by $8^{2}$ by adding the exponents.

Tap for more steps...

Step 11.5.1

Multiply $8$ by $8^{2}$ .

Tap for more steps...

Step 11.5.1.1

Raise $8$ to the power of $1$ .

$\sqrt{\frac{127}{8 + 8^{1} \cdot 8^{2}}}$

Step 11.5.1.2

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

$\sqrt{\frac{127}{8 + 8^{1 + 2}}}$

Step 11.5.2

Add $1$ and $2$ .

$\sqrt{\frac{127}{8 + 8^{3}}}$

Step 11.6

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

$\sqrt{\frac{127}{8 + 512}}$

Step 11.7

Add $8$ and $512$ .

$\sqrt{\frac{127}{520}}$

Step 11.8

Rewrite $\sqrt{\frac{127}{520}}$ as $\frac{\sqrt{127}}{\sqrt{520}}$ .

$\frac{\sqrt{127}}{\sqrt{520}}$

Step 11.9

Simplify the denominator.

Tap for more steps...

Step 11.9.1

Rewrite $520$ as $2^{2} \cdot 130$ .

Tap for more steps...

Step 11.9.1.1

Factor $4$ out of $520$ .

$\frac{\sqrt{127}}{\sqrt{4 (130)}}$

Step 11.9.1.2

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

$\frac{\sqrt{127}}{\sqrt{2^{2} \cdot 130}}$

Step 11.9.2

Pull terms out from under the radical.

$\frac{\sqrt{127}}{2 \sqrt{130}}$

Step 11.10

Multiply $\frac{\sqrt{127}}{2 \sqrt{130}}$ by $\frac{\sqrt{130}}{\sqrt{130}}$ .

$\frac{\sqrt{127}}{2 \sqrt{130}} \cdot \frac{\sqrt{130}}{\sqrt{130}}$

Step 11.11

Combine and simplify the denominator.

Tap for more steps...

Step 11.11.1

Multiply $\frac{\sqrt{127}}{2 \sqrt{130}}$ by $\frac{\sqrt{130}}{\sqrt{130}}$ .

$\frac{\sqrt{127} \sqrt{130}}{2 \sqrt{130} \sqrt{130}}$

Step 11.11.2

Move $\sqrt{130}$ .

$\frac{\sqrt{127} \sqrt{130}}{2 (\sqrt{130} \sqrt{130})}$

Step 11.11.3

Raise $\sqrt{130}$ to the power of $1$ .

$\frac{\sqrt{127} \sqrt{130}}{2 ({\sqrt{130}}^{1} \sqrt{130})}$

Step 11.11.4

Raise $\sqrt{130}$ to the power of $1$ .

$\frac{\sqrt{127} \sqrt{130}}{2 ({\sqrt{130}}^{1} {\sqrt{130}}^{1})}$

Step 11.11.5

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

$\frac{\sqrt{127} \sqrt{130}}{2 {\sqrt{130}}^{1 + 1}}$

Step 11.11.6

Add $1$ and $1$ .

$\frac{\sqrt{127} \sqrt{130}}{2 {\sqrt{130}}^{2}}$

Step 11.11.7

Rewrite ${\sqrt{130}}^{2}$ as $130$ .

Tap for more steps...

Step 11.11.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{130}$ as $130^{\frac{1}{2}}$ .

$\frac{\sqrt{127} \sqrt{130}}{2 {(130^{\frac{1}{2}})}^{2}}$

Step 11.11.7.2

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

$\frac{\sqrt{127} \sqrt{130}}{2 \cdot 130^{\frac{1}{2} \cdot 2}}$

Step 11.11.7.3

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

$\frac{\sqrt{127} \sqrt{130}}{2 \cdot 130^{\frac{2}{2}}}$

Step 11.11.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.11.7.4.1

Cancel the common factor.

$\frac{\sqrt{127} \sqrt{130}}{2 \cdot 130^{\frac{2}{2}}}$

Step 11.11.7.4.2

Rewrite the expression.

$\frac{\sqrt{127} \sqrt{130}}{2 \cdot 130^{1}}$

Step 11.11.7.5

Evaluate the exponent.

$\frac{\sqrt{127} \sqrt{130}}{2 \cdot 130}$

Step 11.12

Simplify the numerator.

Tap for more steps...

Step 11.12.1

Combine using the product rule for radicals.

$\frac{\sqrt{127 \cdot 130}}{2 \cdot 130}$

Step 11.12.2

Multiply $127$ by $130$ .

$\frac{\sqrt{16510}}{2 \cdot 130}$

Step 11.13

Multiply $2$ by $130$ .

$\frac{\sqrt{16510}}{260}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{16510}}{260}$

Decimal Form:

$0.49419709 \dots$