If I have a function $f(x) = g(x)/h(x)$, where $g$ and $h$ are polynomial functions, and I want:
$ \lim_{x \to k} f(x)$, when $h(k) = 0$
Normally I would factor each one of the functions and cancel the 'zeroes' on the numerator with the ones in the denominator, e.g.:
$$ \lim_{x \to 2} \frac{x^2 + x - 6}{x^2 - 3x + 2} = \lim_{x \to 2} \frac{(x - 2)(x + 3)}{(x - 2)(x + 1)} = \lim_{x \to 2} \frac{x+3}{x+1} = 5/3 $$
My question is, if I can't cancel all the 'zeroes' in the denominator, can I say for sure that the limit does not exist? If not, then how can I find it?
It depends on the multiplicity of the root $k$ :
You should be able to write $g(x) = (x-k)^N*\gamma(x)$ with $\gamma(k)\neq 0$
and $h(x) = (x-k)^M*\lambda(x)$ with $\lambda(k) \neq 0$.
M,N are positive integers (could be 0)
Basically the limit is the following :
0 if $N>M$
$+\infty$ if $N<M$
$\frac{\gamma(k)}{\lambda(k)}$ if $N=M$