How to calculate limits of functions rigorously I am using the following definition of the limit of a function "

Definition:   Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Suppose that $E \subseteq X$. Let $f : E \to Y$ and suppose that $p$ is a limit point of $E$. We say that $$\lim_{x \to p} f(x) = q$$ if there exists a point $q \in Y$ with the property that for all $\epsilon > 0$ there exists a $\delta > 0$ such that $$0 < d_X(x, p) < \delta \implies d_Y((f(x), q) < \epsilon$$ for any $x \in B_{(X, d_x)}(p, \delta)$

Suppose now I had the following function $f(x) : \mathbb{R} \setminus \{0\} \to \mathbb{R}$ defined by $$f(x) = \frac{x^2}{x^4 + x^2}$$ and I wanted to prove $$\lim_{x \to 0} \ f(x) = 1.$$  Question 1: What is the easiest way to prove this?

In attempting to prove this on my own I simplified $f(x)$ to get that $$f(x) = \frac{x^2}{x^4 + x^2} = \frac{1}{x^2+1}$$, thus $$\lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{1}{x^2 + 1}$$
Now I could try and prove it directly using the $\epsilon-\delta$ definition above, but I never see anyone actually calculating limits like this in practice (I'm assuming there's a theorem that allows us to prove the above in a much simpler way). 
Indeed there is a theorem that looks like it could be useful here

Theorem: Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, let $E \subseteq X$ and let $f : E \to Y$ . Assume that $p$ is a limit point of $E$, then $f$ is continuous at $p$ if and only if $$\lim_{x \to p} f(x) = f(p)$$

Now note that the function $g : \mathbb{R} \to \mathbb{R}$ defined by $$g(x) = \frac{1}{x^2+1}$$ is continuous on $\mathbb{R}$ and that $0$ is a limit point of $\mathbb{R}$ hence $$\lim_{x \to 0} g(x) = g(0) = 1$$
and this looks like a very promising result that we could use to prove that $\lim_{x \to 0} f(x) = 1$, however there's one big problem, $f$ is defined on $\mathbb{R} \setminus \{0\}$ and since $0$ is not in the domain of $f$, it follows that $f$ can't be continuous at $0$. So we can't make use of the above theorem directly to prove that limit for $f$.
Question 2: Is there any way I can use the above theorem to prove that $\lim_{x \to 0} \ f(x) = 1$?

Regarding your Question 1, you can do a substitution $x=1/u$ to find $ \lim_{u \rightarrow  \infty }  \frac{u^2}{1+u^2} $ instead.
