What is the value of $\lim_{h\to 0}\frac{1/h-a}{h}$

Question

What is the value of $\lim_{h\to 0}\frac{\frac{1}{h}-a}{h}$?

I believe it does not exist, but how can this be proven? If it exists, how would you solve it?

Also, in general, how would you approach a limit like this?

Answer 1

you could make the substitution $\frac{1}{x} = y$ and consider the limits as $y \to \pm \infty$. $$\frac{\frac{1}{x}-a}{x} = \frac{1}{x^2}-\frac{a}{x} = y^2-ay$$ What can you say about $\lim_{y \to \pm \infty} y^2-ay$ for any real number $a$?

Answer 2

For any number $B>0$, and let $|h|<\frac1{2|a|}$. we have

$$\left|\frac{1/h-a}{h}\right|\ge\left|\frac{1-|ha|}{h^2}\right|\ge\frac{1}{2h^2}>B$$

whenever $|h|<\delta =\min\left(\frac{1}{2|a|},\frac{1}{\sqrt{2B}}\right)$.