Is the way we "compute limits" rigorous? This is a general process question. I'm taking my first analysis class in in the chapters of differentiation and continuity, there were quite a few exercises assigned that essentially come down to "compute such-and-such limit". Now, the definition of a limit is clear but for practical purposes isn't constructive; when I look up answers online, or my professor publishes the suggested answers, invariably what is done is the following:
1)
Use some computational trick to reduce the expression whose limit we want to a polynomial or something trivial (factoring, multiplying by some sort of conjugate expression to eliminate radicals, etc). Alternatively, use L'Hospitals rule.
2) Heuristically assume that $\lim_{t \rightarrow x} t^a = x^a$ (heuristic because we justify it in my experience by saying "polynomials are continuous, so the limit must be thus", though in a former thread a member suggested an excellent proof using logarithms:
How do we prove in general that $\lim_{t \to x} {t^a} = x^a$). 
That being said, this doesn't prove that this is the limit. So my questions are:
1)
Are these tools reliable; that is, do they produce limits in general that can be rigorously proven to be correct (using $\epsilon, \delta$ proofs).
2)
When is it safe in a proof to simply say: "let such-and-such tend to zero", or "Let this value go to infinity"?
I have some intuitive ideas, but I can't seem to codify any sort of solid answer, and I'm hoping someone with a lot more experience and a bird's eye view on the problem (both in terms of mathematical savvy and teaching experience) can shed some light.
Thanks.
 A: The method is rigorous. It is based on the following theorems
$$\lim\limits_{x\to a}(f(x)+g(x))=\lim\limits_{x\to a}f(x)+
\lim\limits_{x\to a}g(x)$$
$$\lim\limits_{x\to a}(f(x)g(x))=(\lim\limits_{x\to a}f(x))
(\lim\limits_{x\to a}g(x))$$
$$\lim\limits_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim\limits_{x\to a}f(x)}{
\lim\limits_{x\to a}g(x)}$$
Assuming that $\lim\limits_{x\to a}f(x)$ and $\lim\limits_{x\to a}g(x)$ exist and are finite (and $\lim\limits_{x\to a}g(x)\neq 0$ in the last case).
The aim of the tricks you mention is to express a the function in a form where application of the above theorems is valid. 
There is of course input of basic limits (eg $\lim\limits_{x\to 0}\frac{\sin x}{x}=1$) which must be proved by some other means. 
A: And there are a number of ways
to prove that polynomials
are continuous.
One way is to prove
that the sum and product
of continuous functions
are continuous
and then prove that
1 and x are continuous.
Another,
specialized for polynomials,
is to use
$x^n-a^n
=(x-a)\sum_{k=0}^{n-1} x^k a^{n-1-k}
$
so
$\begin{array}\\
|x^n-a^n|
&=|x-a||\sum_{k=0}^{n-1} x^k a^{n-1-k}|\\
&\le|x-a|n\max(|x|, |a|)^{n-1}\\
&\le|x-a|n(|a|+1)^{n-1}
\qquad\text{if }|x-a| \le 1\\
\end{array}
$
