Calculus Examples

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

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

Step 1

Find where the expression

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

$\frac{x}{\sqrt{2 x - 1}}$ is undefined.

x \leq \frac{1}{2}

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

Step 2

Since

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

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

\to

$\to$

\infty

$\infty$ as

x

$x$

\to

$\to$

\frac{1}{2}

$\frac{1}{2}$ from the left and

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

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

\to

$\to$

\infty

$\infty$ as

x

$x$

\to

$\to$

\frac{1}{2}

$\frac{1}{2}$ from the right, then

x = \frac{1}{2}

$x = \frac{1}{2}$ is a vertical asymptote.

x = \frac{1}{2}

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

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}

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

No Horizontal Asymptotes

Cannot Find Oblique Asymptotes

Step 6