How to Apply Taylor Series at Infinity for x >> 1?

[Vrbic]

Homework Statement

How to use Taylor series for condition x>>1? For example $f(x)=x\sqrt{1+x^2}(2x^2/3-1)+\ln{(x+\sqrt{1+x^2})}$

Homework Equations

The Attempt at a Solution

I try to derived it and limit to infinity...for example first term $\frac{x^4}{3\sqrt{1+x^2}}$. Limit this to infinity is obviously infinity. Any advice?
Thank you.

 

[mfb]

Well, the function goes to infinity, so every approximation should do the same.
What exactly do you want to calculate or approximate where?

In general, to get good approximations for large x, you can calculate the taylor series of f(1/x) and evaluate it around 0. That won't work with divergent series, however, where you might need a Laurent series.

 

[Vrbic]

Well, the function goes to infinity, so every approximation should do the same.
What exactly do you want to calculate or approximate where?

In general, to get good approximations for large x, you can calculate the taylor series of f(1/x) and evaluate it around 0. That won't work with divergent series, however, where you might need a Laurent series.

Nice trick, I tried it but for all derivative limit to 0 is divergent...for example 1.derivative of my f(1/x) is $-\frac{1+1/x^2}{\pi x^4}$. Yes I would like to examine a behaviour of this function for large x and approximate it for this case.

 

[mfb]

Well you'll need a Laurent series here, as I mentioned. If it stops, then you have a good approximation.