How to compute the limit with sinx

Question

I came across this exercise, which leads to computing the limit of $$\exp\left(\lim_{x\to0^+}\sin x\ln(x+\sin x)\right)$$

When using the Taylor expansion, how can one be able to tell should $\sin x$ be replaced with $x$ or $x-{1 \over6}x^3$ just by looking at it, or should I try each of them?

This might seem to be a easy problem, but I really need some help

Answer 1

Multiply and divide by $x$:

$$\frac{\sin(x)}{x}\cdot x\ln(x + \sin(x))$$

Use the known limits:

$$\lim_{x\to 0} \frac{\sin(x)}{x} = 1$$

$$\lim_{x\to 0} x\ln(\alpha x) = 0 ~~~~~~~ \alpha \in\mathbb{R}$$

And then you will find the final limit is

$$\boxed{\color{red}{e^0 = 1}}$$

P.s.

$\ln(x + \sin(x)) \approx \ln(2x)$ as $x\to 0$

Answer 2

It suffices use a first order approximation

• $\sin x \sim x$

and recall that

• $x\log x\to 0$