Calculus Examples

$\frac{lim_{h \to 0} \sqrt[12]{{(x + h)}^{6}} - \sqrt[12]{x^{6}}}{h}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} \sqrt[12]{{(x + h)}^{6}} - lim_{h \to 0} \sqrt[12]{x^{6}}}{h}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt[12]{lim_{h \to 0} {(x + h)}^{6}} - lim_{h \to 0} \sqrt[12]{x^{6}}}{h}$

Step 1.3

Move the exponent $6$ from ${(x + h)}^{6}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt[12]{{(lim_{h \to 0} x + h)}^{6}} - lim_{h \to 0} \sqrt[12]{x^{6}}}{h}$

Step 1.4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{\sqrt[12]{{(lim_{h \to 0} x + lim_{h \to 0} h)}^{6}} - lim_{h \to 0} \sqrt[12]{x^{6}}}{h}$

Step 1.5

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt[12]{{(x + lim_{h \to 0} h)}^{6}} - lim_{h \to 0} \sqrt[12]{x^{6}}}{h}$

Step 1.6

Evaluate the limit of $\sqrt[12]{x^{6}}$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt[12]{{(x + lim_{h \to 0} h)}^{6}} - \sqrt[12]{x^{6}}}{h}$

Step 2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{\sqrt[12]{{(x + 0)}^{6}} - \sqrt[12]{x^{6}}}{h}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $x$ and $0$ .

$\frac{\sqrt[12]{x^{6}} - \sqrt[12]{x^{6}}}{h}$

Step 3.1.2

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\frac{\sqrt[12]{{(x^{2})}^{3}} - \sqrt[12]{x^{6}}}{h}$

Step 3.1.3

Rewrite $\sqrt[12]{{(x^{2})}^{3}}$ as $\sqrt[4]{\sqrt[3]{{(x^{2})}^{3}}}$ .

$\frac{\sqrt[4]{\sqrt[3]{{(x^{2})}^{3}}} - \sqrt[12]{x^{6}}}{h}$

Step 3.1.4

Pull terms out from under the radical, assuming real numbers.

$\frac{\sqrt[4]{x^{2}} - \sqrt[12]{x^{6}}}{h}$

Step 3.1.5

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$\frac{\sqrt{\sqrt{x^{2}}} - \sqrt[12]{x^{6}}}{h}$

Step 3.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{x} - \sqrt[12]{x^{6}}}{h}$

Step 3.1.7

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\frac{\sqrt{x} - \sqrt[12]{{(x^{2})}^{3}}}{h}$

Step 3.1.8

Rewrite $\sqrt[12]{{(x^{2})}^{3}}$ as $\sqrt[4]{\sqrt[3]{{(x^{2})}^{3}}}$ .

$\frac{\sqrt{x} - \sqrt[4]{\sqrt[3]{{(x^{2})}^{3}}}}{h}$

Step 3.1.9

Pull terms out from under the radical, assuming real numbers.

$\frac{\sqrt{x} - \sqrt[4]{x^{2}}}{h}$

Step 3.1.10

Rewrite $\sqrt[4]{x^{2}}$ as $\sqrt{\sqrt{x^{2}}}$ .

$\frac{\sqrt{x} - \sqrt{\sqrt{x^{2}}}}{h}$

Step 3.1.11

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{x} - \sqrt{x}}{h}$

Step 3.1.12

Subtract $\sqrt{x}$ from $\sqrt{x}$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$