Q.Evaluate $\displaystyle\lim_{x\to\infty}\frac{5x^{2} - 3x + 2}{4x^{2} + x - 7}$.

Divide numerator and denominator by $x^2$. …

Direct substitution gives $\tfrac{\infty}{\infty}$. Divide every term by the denominator's highest power, $x^2$:

As $x\to\infty$, each $\tfrac1{x}$ and $\tfrac1{x^2}$ term $\to0$, leaving $\dfrac{5-0+0}{4+0-0}=\dfrac54$. …

Divide by the denominator's highest power, not the numerator's, and never write the answer as $\tfrac{\infty …