Precalculus Examples

$f (x) = - \frac{1}{5} x^{- \frac{1}{3}}$

Step 1

Find where the expression $- \frac{1}{5 x^{\frac{1}{3}}}$ is undefined.

$x = 0$

Step 2

Evaluate $lim_{x \to \infty} - \frac{1}{5 x^{\frac{1}{3}}}$ to find the horizontal asymptote.

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Step 2.1

Evaluate the limit.

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Step 2.1.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to \infty} \frac{1}{5 x^{\frac{1}{3}}}$

Step 2.1.2

Move the term $\frac{1}{5}$ outside of the limit because it is constant with respect to $x$ .

$- \frac{1}{5} lim_{x \to \infty} \frac{1}{x^{\frac{1}{3}}}$

Step 2.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{\frac{1}{3}}}$ approaches $0$ .

$- \frac{1}{5} \cdot 0$

Step 2.3

Multiply $- \frac{1}{5} \cdot 0$ .

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Step 2.3.1

Multiply $0$ by $- 1$ .

$0 (\frac{1}{5})$

Step 2.3.2

Multiply $0$ by $\frac{1}{5}$ .

$0$

Step 3

List the horizontal asymptotes:

$y = 0$

Step 4

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 5

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

No Oblique Asymptotes

Step 6