# How do you determine one sided limits numerically?

Sep 28, 2014

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

${\lim}_{x \to {0}^{-}} \frac{1}{x} = \frac{1}{{0}^{-}} = - \infty$

1 is divided by a number approaching 0, so the magnitude of the quotient gets larger and larger, which can be represented by $\infty$. When a positive number is divided by a negative number, the resulting number must be negative. Hence, then limit above is $- \infty$.

(Caution: When you have infinite limits, those limts do not exist.)

Here is another similar example.

${\lim}_{x \to - {3}^{+}} \frac{2 x + 1}{x + 3} = \frac{2 \left(- 3\right) + 1}{\left(- {3}^{+}\right) + 3} = \frac{- 5}{{0}^{+}} = - \infty$

If no quantity is approaching zero, then you can just evaluate like a two-sided limit.

${\lim}_{x \to {1}^{-}} \frac{1 - 2 x}{{\left(x + 1\right)}^{2}} = \frac{1 - 2 \left(1\right)}{{\left(1 + 1\right)}^{2}} = - \frac{1}{4}$

I hope that this was helpful.