Trying to prove the real-valued function has a limit. I am trying to show that the function $f:\mathbb{R}^2\to\mathbb{R}$ by the formula
$$f(\mathbf{x}) = \dfrac{x_1x_2^2}{x_1^4+x_2^2}  \textrm{ if }\mathbf{x} \neq \mathbf{0}$$
$$f(\mathbf{0}) = 0$$
has a limit of $0$ as $\mathbf{x}\to\mathbf{0}$. 
My book says I have to start by letting $\epsilon>0$ and find $\delta>0$ such that $||\mathbf{x}||<\delta$ implies $|f(\mathbf{x})|<\epsilon$. 
So I let $\epsilon>0$. $$|f(\mathbf{x})| < \epsilon$$
$$|\dfrac{x_1x_2^2}{x_1^4+x_2^2}|<\epsilon$$
I know I have to get $\delta$ in terms of $\epsilon$ so $\delta$ will be chosen sufficiently corresponding to $\epsilon$. I think I need to somehow use $||\mathbf{x}||<\delta \Longleftrightarrow \sqrt{x_1^2+x_2^2} < \delta$ and relate this equation to $|\dfrac{x_1x_2^2}{x_1^4+x_2^2}|<\epsilon$. I am lost and need help on how I would approach this problem next.

Mario's answer is good, but it is also good to practice $\varepsilon-\delta$ proofs. Let $x:=x_1$ and $y:=x_2$ for easy typing. Fix an arbitrary $\varepsilon>0$. First notice that for any $(x,y)\neq (0,0)$
\begin{align*}
|f(x,y)-0|& = \left| \frac{xy^2}{x^4+y^2}\right | \\
& \leq \frac{|x|y^2}{y^2}~\text{noting that}~ x^4,y^2\geq0~\text{,and assuming that}~y\neq 0 \\
&=|x| = \sqrt{x^2} \\
&\leq \sqrt{x^2+y^2} =\|(x,y)-(0,0)\|.
\end{align*}
If $y=0$ then $|f(x,y)-0|=0<|x|$, so our inequality holds for all $(x,y)\neq (0,0)$. Now choose $\delta=\varepsilon>0$. Then for any $(x,y)\in \mathbb{R^2}$ such that $0<\|(x,y)-(0,0)\|<\delta$ it follows from above that $|f(x,y)-(0,0)|<\varepsilon$. By the definition of a limit the result follows. We have found for any $\varepsilon>0$ a $\delta>0$, such that, for any point within a $\delta$ radius of the point, the difference between the value of the function at the point and the suspected limit is less than $\varepsilon$. 
I think it is important to note that having $f(\bar 0)=0$ is only important if you want to prove that $f$ is continuous at $\bar 0$. The whole point of a limit is that we don't care what happens at the point, only arbitrarily close to the point. 
