In Calculus 1, students are taught to find limits of rational polynomials at infinity by dividing by the polynomial of the highest power in the denominator. However, some students want to evaluate the coefficients whenever the numerator and denominator have the same power. Let me explain what I mean by example.
Let $$f(x) = \frac{5x^2+9x+10}{2x^2+x}.$$ Then to evaluate $\lim_{x \rightarrow \infty} f(x)$, of course, the students should divide everything by $x^2$ and see that the limit goes to $\frac{5}{2}$. However, some students simply say "The power is the same on the top and bottom so this becomes $\lim_{x \rightarrow \infty}\frac{5x^2}{2x^2} = \frac{5}{2}$.
This seems like it may not be a valid shortcut, however, if a student asked me why they could not do it, I do not know if I could come up with a good answer. Is there a more complex example where this line of thinking would not hold up?