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In order to find an oblique asymptote, we need to find some function $\phi(x)=kx+n$ or in other words, to find $k$ and $n$. Finding $n$ is pretty straightforward: $$ \begin{align*} \lim_{x \rightarrow \infty}(f(x)-\phi(x))&=0 \\ \lim_{x \rightarrow \infty}(f(x)-kx-n)&=0 \\ \lim_{x \rightarrow \infty}(f(x)-kx)&=n \end{align*} $$ Of course, in order to find $n$, we need $k$ first: $$ \begin{align*} \lim_{x \rightarrow \infty}(f(x)-\phi(x))&=0 \\ \lim_{x \rightarrow \infty}(f(x)-kx-n)&=0 \ \ \text\ \cdot \frac{1}{x} \\ \lim_{x \rightarrow \infty}\left( \frac{f(x)}{x}-k-\frac{n}{x} \right)&=0 \\ \lim_{x \rightarrow \infty}\left( \frac{f(x)}{x} \right)&=k \end{align*} $$ Where $\frac{n}{x}$ tends towards $0$ and we get the formula for $k$.

Question: Are we allowed to divide the whole equation by $x$ in the second step, and if not, why?

I found other similar questions and answers and another way to derive the formula but my professor showed me this one and was confused about it so I'm just wondering why,what's wrong with it.

Cutthroat
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  • It sounds like your professor showed this to you but you don’t believe it’s correct. Is my understanding correct? – Clayton Apr 23 '22 at 12:43
  • He was the one that was confused by it and ended up looking at it for 5 minutes before ending the class. I only have a doubt at what may have been confusing him (the division by $x$) but I'm not sure why. – Cutthroat Apr 23 '22 at 12:45

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