How can I evaluate this limit using Taylor series?
$$\lim_{x \to +\infty} \frac{x – \sin(x) \log(1+x)}{x^7}$$
I know how to evaluate the limit at $0$, but I don't know how to solve it at infinity. If this can't be solved with Taylor series, is there another "easy" way to solve it? (I know you can apply L'Hospital $7$ times, but…)
Best Answer
We know that $\log(1+x)\sim_\infty \log x$ and $\log x=_\infty o(x)$ and $|\sin x|\le 1$ so $$\lim_{x\to\infty}\frac{\sin x\log (1+x)}{x}=0$$
Can you take it from here?