Limit Question, and Existence

Question

I have the following limits let say in this scenario, it's ok if you have one that is a counter-example. The scenario is as follows: \begin{equation} \lim_{x\to c}(\frac{f(x)}{g(x)})\end{equation} I have the following limit laws: \begin{align}\lim_{x\to c}f(x)&=b\\ \lim_{x\to c}g(x)&=a \end{align} Bare with me the limit laws are only valid if the individual limits exist. However, if lets say that $\lim_\limits{x\to c}(\frac{f(x)}{g(x)})$ Just to make it a particular example I set $f(x)=x-6$, and $g(x)=x$ For simplicity I choose my $c=0$. I get that individual limits need to exist, and the overall limit does not exist. \begin{equation} \lim_{x \to 0} f(x) = -6 \end{equation} \begin{equation}\lim_{x\to 0}g(x)=0 \end{equation}Is this how the limit laws work does the overall composite limit have to be defined or can it be undefined at the end, or solely does the individual limits have to exist.

Answer 1

I am converting my comments (to the question) into an answer as requested by the asker. Comments will be deleted to remove redundancy.

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The goal of limit laws is to figure out limit of complicated expressions (which are obtained by applying arithmetic operations on simpler expressions) given some information about the simpler parts of the overall expression.

For the current question the law has hypotheses that limits of $f(x),g(x)$ exist as $x\to c$ and the limit of g(x) is non-zero and the conclusion is that the limit of $f(x)/g(x)$ also exists and its value is the quotient of limits of $f(x)$ and $g(x)$.

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The main confusion with limit laws is that beginners hardly look at the exact hypotheses and conclusions of the limit laws. The goal for them is to get the answer and not look at fine print. Truth is that you can't ignore fine print in analysis/calculus.