Limits and finding a constant value 'k'

[Cpie05]

Stick with me here, I don't know how to use something to add an equation in here!

Homework Statement

Find a value of the constant k such that the limit exists:

lim x->4 (x^2 - k^2) / (x-4)

The Attempt at a Solution

I KNOW the solution is the limit will exist iff k = -4 and k = 4.

My problem is - does the numerator have to cancel out the denominator in order for a limit to exist? Or is that just the case here?

Cause the solution is:

(x-k)(x+k)/(x-4)

So the only way for the denominator (x-4) to cancel out would be if k = -4 or 4... so I'm just wondering if that's a general rule?

Cheers
C.

 

[Dick]

The rule is that if the denominator goes to zero, the only way you can have a limit is if the numerator also goes to zero. For reasons that should be obvious if you think about how division works.

 

[Hurkyl]

Go back and review the limit theorems. Pay particular attention to the details on the one for division; a surprising number of people completely ignore them, and then have trouble dealing with limits that have a division in them.

By applying the fact the limit exists, you should be able to determine something about the thing of which you're taking the limit.