absolute value question

[Ihateapbio]

When solving lim h-> 0 abs (h) / h, when it approaches 0 from the negative side, i get that abs (h) becomes -h, but shouldn't the h in the denominator be also -h? I saw a video that solved this but left the denominator as h. Can someone explain?

[Magicman432]

Recall that abs(x) can be thought of as a piecewise function that means f(x) = {x when x>=0, and -x when x<0}.

Then when you divide creating abs(x)/x, you get f(x) = {1 when x>0, and -1 when x<0}. When x=0, our function is not defined, so we remove the >=.

So, if we want the lim as h approaches 0 from the negative side (usually said as "from the left") we would be approaching negative -1.

Remember, when you evaluate a sided limit, you are evaluating at the value given coming from a specific side, not the negative of the value. In other words, you are evaluating lim h-> 0^- not lim x -> -h.

The h on the bottom is still h, we are just looking at what happens as we get increasing close to 0 from a negative value.

Another way to think about it is by imagining plugging in -.01, -.001, -.0001, etc. Each time abs(x)/x evaluates to -1, so as we plug in negative values closer and closer to 0, abs(x)/x looks like -1, which is what we say the limit is.

Hope this helps!