I've been trying to get the limit of a function, but I don't know how.
- The function is $\displaystyle{10^{n}\left(1 - \mathrm{e}^{\mathrm{i}t/10^{\,n}}\,\right)}$ and the solution says this converges to $-\mathrm{i}t$ as $n \to \infty$.
- The solution also told me to make use of the Euler's formula. I have no clue how they got to $-\mathrm{i}t$.
- Whenever I try to write $\mathrm{e}^{\mathrm{i}t/10^{n}} = \cos\left(t/10^n\right) + \mathrm{i}\,\sin\left(t/10^{n}\right)$ my function converges to $0$.
What did I do wrong ?.
Thank you!
As per the comment section, make the substitution $u = \frac{t}{10^n}$ so that you are computing the limit $$\lim_{u \to 0} \frac{t\left(1 - \cos u - i\sin u\right)}{u} = \lim_{u \to 0} \frac{t(1-\cos u)}{u} - \lim_{u\to 0} \frac{it \sin u}{u}$$
Now making use of the standard limits $\lim_{x\to 0}\frac{1 - \cos x}{x}= 0$ and $\lim_{x\to 0} \frac{\sin x}{x} = 1$ we have
$$\lim_{u \to 0} \frac{t(1-\cos u)}{u} - \lim_{u\to 0} \frac{it \sin u}{u} = t \cdot 0 - it \cdot 1 = -it$$