Limits of rational functions at infinity

Idea. For large $x$, $3x^2$ is much bigger than $x$, and $2x^2$ is much bigger than $5$. We make this visible by dividing everything by the highest power of $x$.

Computation. For $x\neq0$,
$$\frac{3x^2+x}{2x^2-5}=\frac{3+\frac1x}{2-\frac5{x^2}}.$$
As $x\to+\infty$ we have $\frac1x\to0$ and $\frac5{x^2}\to0$, so the numerator tends to $3$ and the denominator to $2$. Hence the limit is $\dfrac32$.

Remark. This recipe works for every rational function. If the numerator has the smaller degree, the limit is $0$. If the degrees are equal, the limit is the ratio of the leading coefficients (here $\frac32$). If the numerator has the larger degree, the quotient tends to $\pm\infty$.