Probably silly question on notation

Question

I just wanted to ask, since $\infty$ is not a number, if $\lim_{x \to a}f(x) \to \infty$ and also $\lim_{x \to b}f(x) \to \infty$, can we write $\lim_{x \to a}f(x)$= $\lim_{x \to b}f(x)$?

Answer 1

So, I am concluding this question.

• Previously, I used to write $\lim_{x\to a} f(x) =c$ when c was finite, but $\lim_{x\to a} f(x)\rightarrow c$ when c tends to $\pm\infty$. For eg.,I’d write $\lim_{x\to 0}\frac{sinx}{x}=1$ but $\lim_{x\to 0}\frac{1}{x}\rightarrow \infty$.
• Now, I have learned that I must use the notation $\lim_{x\to 0}\frac{1}{x}=\infty$, and not $\lim_{x\to 0}\frac{1}{x}\rightarrow \infty$. Please correct me if I have made some mistakes. Thank you.

Answer 2

Such equation may be accepted in some context (for example stating something like $f(x)\leq g(x)\leq h(x)$ and $\lim f(x) = \lim h(x) \implies \lim f(x)= \lim g(x)$) but we need to be careful to avoid mistake by treating such equation as usual equation, for example subtracting two $\infty$s.

I make a remark about a related notion. $\sin x = x+\omicron(x)\ (x→0)$ and $\tan(x)=x+\omicron(x)\ (x→0)$, but obviously $\sin x \neq \tan(x)$.