# Choosing the right order for calculating limits using Taylor expansion

I have been asked to calculate this limit : $\lim_{x\to 0} \frac{e^x + \sin(x) -\cos(x) - 2x}{x - \log(1+x)}$

I have chosen the third order a bit naively thinking that the bigger my order is the more precise my answer will be. I will not write the details of my calculus but after some operations I landed on this limit : $\lim_{x\to 0} \frac{x^2 + x^3\epsilon(x) }{\frac{1}{2}x^2 - \frac{1}{3}x^3 + x^3\epsilon(x^3)}$ and since the $\epsilon$ tends to zero the limite becomes $\lim_{x\to 0} \frac{x^2 }{\frac{1}{2}x^2 - \frac{1}{3}x^3}$ which gives me 2. I've checked the answer and this is the good one but I saw in the corrective that the professor only used the second order expansion so my question is :

Is there a method\trick to immediately know which order you should choose when you notice that you have to use Taylor expansion to compute a limit ?

Thanks by advance !

## 1 Answer

Start with the denominator. Once you know what the leading power is there, you know how far you have to expand the numerator.