Evaluating the limit using Taylor Series We're asked to find the following limit by using Taylor expansions $$\lim_{x\to{}0}\frac{e^{3x}-\sin(x)-\cos(x)+\ln(1-2x)}{-1+\cos(5x)}$$
My Attempt:
Expressing $e^{3x}$, $\sin(x)$, $\cos(x)$, $\ln(1-2x)$ and $\cos(5x)$ in their respective taylor expansions yielded the following monstrous fraction, https://imgur.com/a/xGyfIyL (Picture size too big to be uploaded here for some reason, plus fraction too large to be expressed in the space given :/) But anyways, I can't seem to factorize this thing and evaluate the limit as $x\to{}0$, any help would be appreciated.

We need only quote the numerator and denominator up to $x^2$ terms: $$\lim_{x\to 0}\frac{1+3x+\color{blue}{9x^2/2}-x-1+\color{blue}{x^2/2}-2x\color{blue}{-2x^2}+O(x^3)}{-1+1\color{blue}{-25x^2/2}+O(x^3)}.$$You'll find only $x^2$ terms survive in each.
