I wonder how to compute the limit of error term in Taylor polynomial of exponential functions (i.e. $f(x)=e^{3x}-1$).

First, I found the Taylor polynomial of $f(x)$, which is $$T_n f(x)=\frac{3^1}{1!}x^1 + \frac{3^2}{2!}x^2 + \frac{3^3}{3!}x^{3} + \cdots + \frac{3^{n}}{n!}x^{n}.$$

Then, I used the remainder theorem to express the error term: $R_nf(x) = \dfrac{3^{n+1}e^{3\xi}}{(n+1)!}x^{n+1}$, where $0 < \xi < x$ ($x>0$) or $ x < \xi < 0$ $(x<0)$.

I know $\lim_{x\rightarrow \infty} \frac{3^{n+1}}{(n+1)!}x^{n+1} = 0$, but I have no ideas what shall I do next, when $e^{3x}$ involved.

Any suggestions?



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