Calculus Examples

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

Step 1

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

$\frac{lim_{x \to 8} x \cdot lim_{x \to 8} \sqrt{x}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 2

Move the limit under the radical sign.

$\frac{lim_{x \to 8} x \cdot \sqrt{lim_{x \to 8} x}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 3

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

Tap for more steps...

Step 3.1

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

$\frac{8 \cdot \sqrt{lim_{x \to 8} x}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 3.2

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

$\frac{8 \cdot \sqrt{8}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 4.1.1.1

Factor $4$ out of $8$ .

$\frac{8 \sqrt{4 (2)}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 4.1.1.2

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

$\frac{8 \sqrt{2^{2} \cdot 2}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 4.1.2

Pull terms out from under the radical.

$\frac{8 \cdot 2 \sqrt{2}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 4.1.3

Multiply $8$ by $2$ .

$\frac{16 \sqrt{2}}{\sqrt{x^{3} + 2 x^{2}}}$

Step 4.2

Simplify the denominator.

Tap for more steps...

Step 4.2.1

Factor $x^{2}$ out of $x^{3} + 2 x^{2}$ .

Tap for more steps...

Step 4.2.1.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{16 \sqrt{2}}{\sqrt{x^{2} x + 2 x^{2}}}$

Step 4.2.1.2

Factor $x^{2}$ out of $2 x^{2}$ .

$\frac{16 \sqrt{2}}{\sqrt{x^{2} x + x^{2} \cdot 2}}$

Step 4.2.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 2$ .

$\frac{16 \sqrt{2}}{\sqrt{x^{2} (x + 2)}}$

Step 4.2.2

Pull terms out from under the radical.

$\frac{16 \sqrt{2}}{x \sqrt{x + 2}}$

Step 4.3

Multiply $\frac{16 \sqrt{2}}{x \sqrt{x + 2}}$ by $\frac{\sqrt{x + 2}}{\sqrt{x + 2}}$ .

$\frac{16 \sqrt{2}}{x \sqrt{x + 2}} \cdot \frac{\sqrt{x + 2}}{\sqrt{x + 2}}$

Step 4.4

Combine and simplify the denominator.

Tap for more steps...

Step 4.4.1

Multiply $\frac{16 \sqrt{2}}{x \sqrt{x + 2}}$ by $\frac{\sqrt{x + 2}}{\sqrt{x + 2}}$ .

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x \sqrt{x + 2} \sqrt{x + 2}}$

Step 4.4.2

Move $\sqrt{x + 2}$ .

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x (\sqrt{x + 2} \sqrt{x + 2})}$

Step 4.4.3

Raise $\sqrt{x + 2}$ to the power of $1$ .

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x ({\sqrt{x + 2}}^{1} \sqrt{x + 2})}$

Step 4.4.4

Raise $\sqrt{x + 2}$ to the power of $1$ .

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x ({\sqrt{x + 2}}^{1} {\sqrt{x + 2}}^{1})}$

Step 4.4.5

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

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x {\sqrt{x + 2}}^{1 + 1}}$

Step 4.4.6

Add $1$ and $1$ .

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x {\sqrt{x + 2}}^{2}}$

Step 4.4.7

Rewrite ${\sqrt{x + 2}}^{2}$ as $x + 2$ .

Tap for more steps...

Step 4.4.7.1

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

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x {({(x + 2)}^{\frac{1}{2}})}^{2}}$

Step 4.4.7.2

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

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x {(x + 2)}^{\frac{1}{2} \cdot 2}}$

Step 4.4.7.3

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

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x {(x + 2)}^{\frac{2}{2}}}$

Step 4.4.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.7.4.1

Cancel the common factor.

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x {(x + 2)}^{\frac{2}{2}}}$

Step 4.4.7.4.2

Rewrite the expression.

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x {(x + 2)}^{1}}$

Step 4.4.7.5

Simplify.

$\frac{16 \sqrt{2} \sqrt{x + 2}}{x (x + 2)}$

Step 4.5

Combine using the product rule for radicals.

$\frac{16 \sqrt{(x + 2) \cdot 2}}{x (x + 2)}$

Step 4.6

Move $2$ to the left of $x + 2$ .

$\frac{16 \sqrt{2 (x + 2)}}{x (x + 2)}$