$\lim\limits_{x\to \infty} \dfrac{x^4+3x^3-2x+5}{3x^4+5x^2-13x}$

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When determining limits at infinity, only the highest power terms in the numerator and denominator are relevant.

$$\lim\limits_{x\to \infty} \dfrac{x^4+3x^3-2x+5}{3x^4+5x^2-13x} \approx \lim\limits_{x\to \infty} \dfrac{x^4}{3x^4}$$ $$= \lim\limits_{x\to \infty} \dfrac{1}{3}$$ $$= \boxed{\frac{1}{3}}$$

We can divide the numerator and denominator by $x^4$.

$$\lim\limits_{x\to\infty} \frac{1+\frac{3}{x}-\frac{2}{x^3}+\frac{5}{x^4}}{3+\frac{5}{x^2}-\frac{13}{x^3}}$$

Recall that as $x$ approaches large values, fractions in the form of $\dfrac{c}{x^r}$ approach 0.

$$= \frac{1+0+0+0}{3+0+0}$$ $$= \boxed{\frac{1}{3}}$$