Calculus Examples

$y = \frac{20 x}{{(x^{4} + 1)}^{\frac{1}{4}}}$

Step 1

Find where the expression $\frac{20 x}{{(x^{4} + 1)}^{\frac{1}{4}}}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Evaluate $lim_{x \to \infty} \frac{20 x}{{(x^{4} + 1)}^{\frac{1}{4}}}$ to find the horizontal asymptote.

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Step 3.1

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$20 lim_{x \to \infty} \frac{x}{{(x^{4} + 1)}^{\frac{1}{4}}}$

Step 3.2

Rewrite ${(x^{4} + 1)}^{\frac{1}{4}}$ as $\sqrt[4]{x^{4} + 1}$ .

$20 lim_{x \to \infty} \frac{x}{\sqrt[4]{x^{4} + 1}}$

Step 3.3

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x = \sqrt[4]{x^{4}}$ .

$20 lim_{x \to \infty} \frac{\frac{x}{x}}{\sqrt[4]{\frac{x^{4}}{x^{4}} + \frac{1}{x^{4}}}}$

Step 3.4

Evaluate the limit.

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Step 3.4.1

Cancel the common factor of $x$ .

$20 lim_{x \to \infty} \frac{1}{\sqrt[4]{\frac{x^{4}}{x^{4}} + \frac{1}{x^{4}}}}$

Step 3.4.2

Cancel the common factor of $x^{4}$ .

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Step 3.4.2.1

Cancel the common factor.

$20 lim_{x \to \infty} \frac{1}{\sqrt[4]{\frac{x^{4}}{x^{4}} + \frac{1}{x^{4}}}}$

Step 3.4.2.2

Rewrite the expression.

$20 lim_{x \to \infty} \frac{1}{\sqrt[4]{1 + \frac{1}{x^{4}}}}$

Step 3.4.3

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

$20 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} \sqrt[4]{1 + \frac{1}{x^{4}}}}$

Step 3.4.4

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

$20 \frac{1}{lim_{x \to \infty} \sqrt[4]{1 + \frac{1}{x^{4}}}}$

Step 3.4.5

Move the limit under the radical sign.

$20 \frac{1}{\sqrt[4]{lim_{x \to \infty} 1 + \frac{1}{x^{4}}}}$

Step 3.4.6

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

$20 \frac{1}{\sqrt[4]{lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{1}{x^{4}}}}$

Step 3.4.7

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

$20 \frac{1}{\sqrt[4]{1 + lim_{x \to \infty} \frac{1}{x^{4}}}}$

Step 3.5

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{4}}$ approaches $0$ .

$20 \frac{1}{\sqrt[4]{1 + 0}}$

Step 3.6

Simplify the answer.

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Step 3.6.1

Simplify the denominator.

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Step 3.6.1.1

Add $1$ and $0$ .

$20 \frac{1}{\sqrt[4]{1}}$

Step 3.6.1.2

Any root of $1$ is $1$ .

$20 (\frac{1}{1})$

Step 3.6.2

Cancel the common factor of $1$ .

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Step 3.6.2.1

Cancel the common factor.

$20 (\frac{1}{1})$

Step 3.6.2.2

Rewrite the expression.

$20 \cdot 1$

Step 3.6.3

Multiply $20$ by $1$ .

$20$

Step 4

List the horizontal asymptotes:

$y = 20$

Step 5

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 6

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 20$

No Oblique Asymptotes

Step 7