## Valdas 3 years ago Finding a limit

1. Valdas

$\lim_{x \rightarrow 0} \frac{ \sqrt[5]{1+2x}-1 }{ \sin x }$

2. myko

try lhopital

3. Valdas

sorry, I forgot to mention that we haven't learnt about lhopital

4. myko

nvm

5. myko

you learned Taylors series?

6. Valdas

Nope

7. RajshikharGupta

expand the root term by binomial theorem

8. RajshikharGupta

then there will be term like 10x/sinx and x/sinx=1 for given condition so ans =10

9. RajshikharGupta

while opening with binomial neglect higher degree terms

10. Valdas

well, the answer should be 0.4 . And what do you mean by expanding the root term by binomial theorem? Do you mean, that I should raise the fraction by the 5th degree and then use the binomial theorem and numerator?

11. RajshikharGupta

no see it i have shown|dw:1353844296253:dw|

12. RajshikharGupta

and i have done mistake while doing this it is 2/5=0.4 so sorry for that

13. Valdas

Could you explain how did you get $\sqrt[5]{1+2n}=1+\frac{ 2 }{ 5 }n + ....?$

14. RajshikharGupta

it is binomial theorem u will learn this in algebra in high school it is (1+x)^n=1+(nC1)x+(nC2)x^2+(nC3)x^3 and so on .....

15. Valdas

Yes, I know the binomial theorem, but how does it apply to roots? I thought that the power of polynomial it is raised to must be an integer to apply this theorem.

16. RajshikharGupta

no it can be applied on roots if x is very small and it can even for any fractional powers for same case it quite valid approximation tool in maths and physics u can conform it with ur teachers

17. Valdas

Perhaps anyone can think of other way to find the limit? Every problem until this one required some sort of quite simple algebraic manipulation.

18. RajshikharGupta

actually it is one of the shortest methods but indeed u can initiate in ur problem by factorizing the numerator term

19. pasta

Binomial theorem just makes work hard ,learn l hopitals rule it is easy.LOOK IF after substituting the limit you get 0/0 differentiate the numenartor and denominator independently the find the limit three steps

20. Valdas

I can't use l'hoptial's rule to find the limit

21. pasta

i do not get you.Do you mean ,you are not allowed to use l hopitals rule or you can not evaluate.

22. Valdas

We haven't studied about that rule yet

23. pasta

do u want to learn it?

24. Valdas

This problem can be solved without use of l'hopitals rule, I want to find out how. Perhaps there is some simple algebraic trick which could be used to find the limit or something like that

25. shubhamsrg

@RajshikharGupta 's method seems to be the shortest to me,

26. pasta

ok ,I SEE.I have two solutions so far ,binomial and l'hopitals.But looks like there are one's and may be try following the trigonometry side or otherwise am still checking

27. shubhamsrg

are you asking how do we get (1+x)^n = 1+xn (given |x|<1 ) ?

28. Valdas

Yeah, I don't underst that part

29. shubhamsrg

hmm,,heard of taylor series expansion ?

30. Valdas

No

31. shubhamsrg

then you might just wanna mug up this formulla : |dw:1353848320044:dw| you'll get to know how we get this when you learn taylor series..

32. shubhamsrg

make that substitution in your question.. you'll reach the ans directly then..