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Limits question from my book in Latex

Actual Question with solution in my book

First image is the question from my textbook in Latex form. Second image shows the actual question from my book along with the solution.

What i am actually confused about is that how did they open the power 1/3 and got the following result on each step.

Is there anyone who can help me more to understand each step of this question? Or help me understand a new way to do this question? Or something like this.

Help will be highly appreciated.

Regards, Ballu Miaa

Ballu Miaa
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1 Answers1

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They did the Taylor expansion of it:

$$(1+h)^{1/3}=1+\frac{1}{3}h+o(h^2)$$ $$(1-h)^{1/3}=1-\frac{1}{3}h+o(h^2)$$

Then, they use the fact that

$$\lim_{h\rightarrow0}\frac{o(h^2)}{h}=0$$

So they just keep the first power (the $1$s cancel)

MyUserIsThis
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  • Looks good. Actually solved my question. But Taylor expansion is used to do what actually? It is a way to expand an algebraic expression? Also there were no other ways to do this kind of question?

    Thanks a lot man. Much Love. You're a saviour.

     – Ballu Miaa Nov 23 '13 at 11:58
  • Also can you tell me how to use Latex in a post? The way you write the formula over here above you know. – Ballu Miaa Nov 23 '13 at 12:05
  • @BalluMiaa, there are other ways to solve such problem, see http://math.stackexchange.com/questions/569230/cant-find-the-limit-of-the-following – lab bhattacharjee Nov 23 '13 at 12:07
  • Ohh thats a good way too. Thanks a lot Bhattacharjee. I will look at both of these methods for future references. – Ballu Miaa Nov 23 '13 at 12:13
  • @BalluMiaa. Taylor expansion is a very good tool to approximate, even very accurately, very complex functions around a given point. Look at the Wiki page mentioned above. – Claude Leibovici Nov 23 '13 at 13:07
  • @ClaudeLeibovici - Thanks a lot. Well i been away from Maths since some years and started it again to pass my uncleared exams. Therefore its hard for me grasp the language used in that wiki article. Well you helped me enough to understand it. – Ballu Miaa Nov 23 '13 at 14:01
  • @BalluMiaa. Happy and glad of being able to help you. Cheers. – Claude Leibovici Nov 24 '13 at 14:33