Let $f,g \in L^2$, find the limit $\lim_{t \to \infty}\frac{\|f+tg\|_2-\|f\|_2}{t}$

Question

Let $f,g \in L^2$, find the limit $\lim_{t \to \infty}\frac{\|f+tg\|_2-\|f\|_2}{t}$. My guess is that the limit is $\|g\|_2$ as applying Minkowski gives us this as an upper bound for the limit. I was not successful however, in showing that this is in fact the limit.

Answer 1

You're on the right track. I'm going to do a little trick here to make this more manageable. Let's assume $t>0$ since it's trending to $+\infty$ anyway, so

$$\lim_{t\to\infty} \frac{\|f+tg\|-\|f\|}{t} = \lim_{t\to\infty} \frac{t\|t^{-1}f+g\|-t\|t^{-1}f\|}{t} = \lim_{t\to\infty} (\|t^{-1}f+g\|-\|t^{-1}f\|).$$

A simple rewrite of $r = t^{-1}$ will make this limit clearer. Can you take it from here? Inherent in the final step is that norms are continuous functions.

Answer 2

$$\lim_{t\to\infty}\frac{||f+tg||-||f||}{t} = \lim_{t\to\infty}\left(\left|\left|\frac{f}{t}+g\right|\right|-\frac{||f||}{t}\right)=||g||$$