How to get $\lim_{t \to 0} \frac{\left \|a + t\frac{a}{\|a\|} \right\| - \|a\|}{t} = \lim_{t \to 0} \frac{(\|a\|+t) - \|a\|}{t}$?

Question

I'm watching Professor Shifrin's lectures here, at which he mentioned:

$$\lim_{t \to 0} \frac{\left \|a + t\dfrac{a}{\|a\|} \right\| - \|a\|}{t} = \lim_{t \to 0} \frac{(\|a\|+t) - \|a\|}{t}$$

In my computation, $\left \|a + t\dfrac{a}{\|a\|} \right\| = \|a\| \cdot |1+t/\|a\||$. I could not simplify more. Could you please elaborate on how to get the result?

Answer 1

There's no super tricky calculation involved, actually; all you need is to add the fraction and the $1$ to see what's happening:

$$ \left \|a + t\dfrac{a}{\|a\|} \right\| = \|a\| \cdot \left|1+ \frac{t}{\|a\|} \right| = \|a\| \cdot \left|\frac{\|a\| + t}{\|a\|} \right| = \left|\|a\| + t \right| = \|a\| + t . $$

Then, what you wanted becomes clear!