Strategy to compute limit of a complex function

Question

Does a "standard" procedure to compute limits of the complex functions exist? I know that this question is generic. I expect a generic reply.

Thank you so much in advance.

Answer 1

When we are dealing with indeterminate form of complicated expression, the standard and more effective method to calculate the limit is of course Taylor's series expansion.

In that way we can reduce any term in polynomial form and the limit becomes easy do solve. The difficult is to select the correct (i.e. minimum) order of expansion and to deal with the remainder terms (usually in little-o or big-O notation).

For example by Taylor's series, with some practice, it is easy to see that

$$\lim_{x\to 0} \frac{ \sin^2 \left( \arctan (\log (1+\sin x) \right) -1+\cos(\tan x)} {\arcsin^4(\sin(\log (1-\tan(x^2))))+e^{\sin^2x}-1}=\frac12$$

indeed it correspons to

$$\lim_{x\to 0} \frac{ x^2-1+\frac12x^2+o(x^2)} {x^8+1+x^2-1+o(x^2)}= \lim_{x\to 0} \frac{ \frac12x^2+o(x^2)} {x^2+o(x^2)}=\frac12$$