Existence of limit of a function with a parameter

[AwesomeTrains]

Homework Statement

For what values of a, from the reals, does the limit exist?
$lim_{x\rightarrow2} (\frac{1}{2-x}-\frac{a}{4-x^{2}})$

Homework Equations

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

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.

 

[LCKurtz]

Homework Statement

For what values of a, from the reals, does the limit exist?
$lim_{x\rightarrow2} (\frac{1}{2-x}-\frac{a}{4-x^{2}})$

Homework Equations

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

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.

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.