Consider $\displaystyle \lim_{x \to -6} \frac{2x+12}{|x+6|}$.

I thought the best way to approach this was to consider the two sides of the limit seperately.

$\displaystyle \lim_{x \to -6^+} \frac{2x+12}{|x+6|}$

$\displaystyle = \lim_{x \to -6^+} \frac{2x+12}{6-x}$ because x < 0.

Now the limit will evaluate to 0. I had to check numerically to discover the limits on both sides of -6 are -2 and 2 respectively, so there is no limit.

I had a hunch from the fact that the limit seemed to be equal to 0.

But how can I, without tedious checking values,

a) Establish whether there is a limit or not

b) The value of the limit if it exists?

If I carried on applying algebraic rules and limit laws to this problem I would get the answer 0, and be none the wiser.