# What is the limit of f(x) as x approaches 0?

Jul 24, 2015

It depends on your function really.

#### Explanation:

You can have various types of functions and various behaviours as they approach zero;
for example:

1] $f \left(x\right) = \frac{1}{x}$ is very strange, because if you try to get near zero from the right (see the little $+$ sign over the zero):
${\lim}_{x \to {0}^{+}} \frac{1}{x} = + \infty$ this means that the value of your function as you approach zero becomes enormous (try using: $x = 0.01 \mathmr{and} x = 0.0001$).

If you try to get near zero from the left (see the little $-$ sign over the zero):
${\lim}_{x \to {0}^{-}} \frac{1}{x} = - \infty$ this means that the value of your function as you approach zero becomes enormous but negative (try using: $x = - 0.01 \mathmr{and} x = - 0.0001$).

2] $f \left(x\right) = 3 x + 1$ as you approach zero from the right or left your function tends to $1$!
${\lim}_{x \to 0} \left(3 x + 1\right) = 1$

Basically, as a general rule, when you have to evaluate a limit for $x \to a$ try first to substitute $a$ into your function and see what happens. If you get something problematic such as $\frac{0}{0} \mathmr{and} \frac{\infty}{\infty} \mathmr{and} \frac{1}{0}$ try to get as near as possible to $a$ and see if you "see" a pattern, a trend...a tendency!