Calculus Examples

$\frac{1}{\sqrt{x}}$

Step 1

Find where the expression $\frac{1}{\sqrt{x}}$ is undefined.

$x \leq 0$

Step 2

Since $\frac{1}{\sqrt{x}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{1}{\sqrt{x}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 3

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

$0$

Step 4

List the horizontal asymptotes:

$y = 0$

Step 5

Use polynomial division to find the oblique asymptotes. Because this expression contains a radical, polynomial division cannot be performed.

Cannot Find Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

Horizontal Asymptotes: $y = 0$

Cannot Find Oblique Asymptotes

Step 7