Calculus Examples

$\frac{lim_{x \to 8} \sqrt[4]{x^{8} - 4 x}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 1

Move the limit under the radical sign.

$\frac{\sqrt[4]{lim_{x \to 8} x^{8} - 4 x}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 2

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

$\frac{\sqrt[4]{lim_{x \to 8} x^{8} - lim_{x \to 8} 4 x}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 3

Move the exponent $8$ from $x^{8}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

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

$\frac{\sqrt[4]{8^{8} - 4 lim_{x \to 8} x}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 5.2

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

$\frac{\sqrt[4]{8^{8} - 4 \cdot 8}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

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

$\frac{\sqrt[4]{16777216 - 4 \cdot 8}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 6.1.2

Multiply $- 4$ by $8$ .

$\frac{\sqrt[4]{16777216 - 32}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 6.1.3

Subtract $32$ from $16777216$ .

$\frac{\sqrt[4]{16777184}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 6.1.4

Rewrite $16777184$ as $2^{4} \cdot 1048574$ .

Tap for more steps...

Step 6.1.4.1

Factor $16$ out of $16777184$ .

$\frac{\sqrt[4]{16 (1048574)}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 6.1.4.2

Rewrite $16$ as $2^{4}$ .

$\frac{\sqrt[4]{2^{4} \cdot 1048574}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 6.1.5

Pull terms out from under the radical.

$\frac{2 \sqrt[4]{1048574}}{2 x^{2} - \sqrt{x^{4} - 2}}$

Step 6.2

Multiply $\frac{2 \sqrt[4]{1048574}}{2 x^{2} - \sqrt{x^{4} - 2}}$ by $\frac{2 x^{2} + \sqrt{x^{4} - 2}}{2 x^{2} + \sqrt{x^{4} - 2}}$ .

$\frac{2 \sqrt[4]{1048574}}{2 x^{2} - \sqrt{x^{4} - 2}} \cdot \frac{2 x^{2} + \sqrt{x^{4} - 2}}{2 x^{2} + \sqrt{x^{4} - 2}}$

Step 6.3

Multiply $\frac{2 \sqrt[4]{1048574}}{2 x^{2} - \sqrt{x^{4} - 2}}$ by $\frac{2 x^{2} + \sqrt{x^{4} - 2}}{2 x^{2} + \sqrt{x^{4} - 2}}$ .

$\frac{2 \sqrt[4]{1048574} (2 x^{2} + \sqrt{x^{4} - 2})}{(2 x^{2} - \sqrt{x^{4} - 2}) (2 x^{2} + \sqrt{x^{4} - 2})}$

Step 6.4

Expand the denominator using the FOIL method.

$\frac{2 \sqrt[4]{1048574} (2 x^{2} + \sqrt{x^{4} - 2})}{4 x^{4} + 2 x^{2} \sqrt{x^{4} - 2} - 2 \sqrt{x^{4} - 2} x^{2} - {\sqrt{x^{4} - 2}}^{2}}$

Step 6.5

Simplify.

$\frac{2 \sqrt[4]{1048574} (2 x^{2} + \sqrt{x^{4} - 2})}{3 x^{4} + 2}$