# Existence of limit of a function with a parameter

1. Jun 14, 2014

### AwesomeTrains

1. The problem statement, all variables and given/known data
For what values of a, from the reals, does the limit exist?
$lim_{x\rightarrow2} (\frac{1}{2-x}-\frac{a}{4-x^{2}})$

2. Relevant equations
I chose a so that the denominator would be one. By putting the fractions together.

3. The attempt at a solution
When a = 4 the denominator of the combined fraction can be reduced to one
=> then the limit is -1/4.

(For $a=x+2$ and $a=x^{2}+x-2$ the denominator is 1 too, but at x=2 all three solutions are equal to 4.)

$\textbf{tl;dr}$
$\textbf{My question: Is 4 the only solution?}$

In the problem statement a is in plural.
Am I missing any solutions?

Any hints are much appreciated.

2. Jun 14, 2014

### LCKurtz

You have the correct answer. To see that there are no other answers, add the two fractions together and see if you can argue that having a finite limit implies a = 4.