Expert: Paul Klarreich - 10/8/2009

Question
I need to find the limit as x approaches 9 from the left side. I am having troubles understand how the absolute value factors in and how to solve without getting a zero in the denominator.
lim_(x->-9) (9 - abs(x))/(9 + x) . I have tried to figure out a way to get rid of the x on the bottom but I dont understand how the abs val of x applies. Any help you can give will be appreciated

Answer
Questioner: Kaitlyn
Country: Canada
Category: Calculus
Private: No
Subject: Calculating limits using the limit laws.
Question: I need to find the limit as x approaches 9 from the left side. I am having troubles understand how the absolute value factors in and how to solve without getting a zero in the denominator.
lim_(x->-9) (9 - abs(x))/(9 + x) . I have tried to figure out a way to get rid of the x on the bottom but I dont understand how the abs val of x applies. Any help you can give will be appreciated
............................
Hi, Kaitlin,

A good thing to do when you don't understand how to use something is to go back to the definition:

abs(x) = x, when x >=0, and THE OPPOSITE OF x, when x < 0.

Then you will see that if you are looking at numbers NEAR x = -9 (that's what limit means) then you want to use  THE OPPOSITE OF x  for  abs(x).

Of course,  "THE OPPOSITE OF x" is normally written  "-x".

So your limit becomes:

lim_(x->-9) (9 - (-x))/(9 + x)

Does that do it for you?