Exercise 2. Compute the following limit$$\lim_{x\to0^+}\frac{2\arctan(x)\sqrt{1-x^2}+\frac{x^3-2x^4}{\ln(2x+2x^2)}-\ln(1+2x+2x^2)}{e^{x-\frac{x^2}2}-\cos(x^2)+\tan(2x^4-x)+x^2e^{-\frac1{x}}}$$
My question is that is there any special formula for this part of question? As you can see, question has a lot of parts that are needed to be solved seperately . I used $(x-x^3/3+x^5/5-...)$ for $\arctan(x)$, and $e^x=1+x+x^2/2!+x^3/3!+...$ and else. But I couldn't find any formula for this section
The middle term in the numerator goes to zero more quickly than $x^3$ because of the division by $\ln x$. It turns out the numerator and denominator are of order $x^3$ so you can ignore it and work on the rest. There is the standard series of $\ln (1+x) \approx x-\frac {x^2}2+\frac {x^3}3-\ldots$
I put the numerator into Alpha and it is about $-\frac {x^3}3$. The denominator is about $-\frac {2x^3}3$ so the limit is $\frac 12$