# How do you find the limit lim_(x->0^-)|x|/x ?

Aug 2, 2014

When dealing with one-sided limits that involve the absolute value of something, the key is to remember that the absolute value function is really a piece-wise function in disguise. It can be broken down into this:

$| x | =$

$x$, when $x \ge 0$
-$x$, when $x < 0$

You can see that no matter what value of $x$ is chosen, it will always return a non-negative number, which is the main use of the absolute value function. This means that to evaluate this one-sided limit, we must figure out which version of this function is appropriate for our question.

Because our limit is approaching $0$ from the negative side, we must use the version of $| x |$ that is $< 0$, which is $- x$. Rewriting our original problem, we have:

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

Now that the absolute value is gone, we can divide the $x$ term and now have:

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

One of the properties of limits is that the limit of a constant is always that constant. If you imagine a constant on a graph, it would be a horizontal line stretching infinitely in both directions, since it stays at the same $y$-value regardless of what the $x$-value does. Since limits deal with finding what value a function "approaches" as it reaches a certain point, the limit of a horizontal line will always be a point along that line, no matter what x-value is chosen. Because of this, we now know:

${\lim}_{x \to {0}^{-}} - 1 = - 1$, Giving us our final answer.