Calculus Examples

$\frac{x}{\sqrt{2 x - 1}}$

Step 1

Find where the expression $\frac{x}{\sqrt{2 x - 1}}$ is undefined.

$x \leq \frac{1}{2}$

Step 2

Since $\frac{x}{\sqrt{2 x - 1}}$ $\to$ $\infty$ as $x$ $\to$ $\frac{1}{2}$ from the left and $\frac{x}{\sqrt{2 x - 1}}$ $\to$ $\infty$ as $x$ $\to$ $\frac{1}{2}$ from the right, then $x = \frac{1}{2}$ is a vertical asymptote.

$x = \frac{1}{2}$

Step 3

Since the limit does not exist, there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 4

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

Cannot Find Oblique Asymptotes

Step 5

This is the set of all asymptotes.

Vertical Asymptotes: $x = \frac{1}{2}$

No Horizontal Asymptotes

Cannot Find Oblique Asymptotes

Step 6