Question

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

Answer 1

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>= 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->0^-)(-x)/x`

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

`lim_(x->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->0^-)-1 = -1`, Giving us our final answer.