Using the $\epsilon-\delta$ definition of the limit, evaluate $\lim_{x \to a} f(x)$, where $f(x) = x^3+5x$.
Attempt
We can use the usual way of constructing the limit: $$\forall \epsilon, \exists \delta \quad 0 < |x-a| < \delta \quad \implies \quad |x^3+5x-(a^3+5a)| < \epsilon.$$ Then we get $|x-a||a^2+x(a+x)+5| < \epsilon$. I imaging now we are going to have to bound $\delta$ by say $\leq 1$ to get $|x-a| < 1$. Then I'm not sure what to do next.