Prove that $(e^{At}−I)/t→A$ as $t→0$, meaning $\|(e^{At}−I)/t−A\|\to 0$ as $t\to0$ for all $A\in C^{n×n}$.

Question

a) Prove that $(e^{At}−I)/t\to A$ as $t\to 0$, meaning $\|(e^{At}−I)/t−A\|\to 0$ as $t\to 0$ for all $A\in C^{n×n}$. Hint: You may use the inequality $\|A^k\|\leq n^{k−1}\|A\|^k$

I've been looking at this question and i can prove it using the Taylor expansion which is quite straightforward:

$$e^{At}=I+At+{(At)^2}/2!+...+{(At)^k}/k!\\ e^{At}-I=At+{(At)^2}/2!+...+{(At)^k}/k!\\ (e^{At}-I)/t=A+{A^2t}/2!+...+{A^kt^{k-1}}/k!\\ \|(e^{At}−I)/t−A\|={A^2t}/2!+...+{A^kt^{k-1}}/k!$$

Therefore as $t→0$, $\|(e^{At}−I)/t−A\|\to 0$

However, this doesn't use the Hint given in the question. Could someone give me a clue as to how to prove the question using the Hint please?

Answer 1

What you have written as Taylor expansion should really be an infinite series. Then you have to justify to pull the limit "$t\to 0$" inside of this series. To do so, use the hint.