Expert: Socrates - 9/3/2006

Question
I am trying to find the limit as x approaches 3 from the left.  I am stumped due to the absolute value in the equation.

f(x){|x-3|/x-3 when x < 3
and  x         when x > 3}

I know since the approach is from the left only the top function matters.  I also realize that no matter what you plug into x the value will always be -1 below zero.  I need to understand how this problem works because I will see it on a test soon.  Thanks for any input you have.

Answer
You are right, the only expression that matters in finding this limit is

|x-3|/(x-3)

When x approaches 3 from the left, x is always less than 3 , so x-3 is always negative.

When the expression in absolute value is negative, you multiply the expression by -1 and drop the absolute value bars.

So this means |x-3| = -(x-3) when x approaches 3 from the left.

We now have

|x-3|/(x-3) = -(x-3)/(x-3) = -1

so the limit as x goes to 3 from the left is -1