Finding a limit

- anonymous

Finding a limit

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- anonymous

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

- anonymous

try l`hopital

- anonymous

sorry, I forgot to mention that we haven't learnt about l`hopital

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## More answers

- anonymous

nvm

- anonymous

you learned Taylors series?

- anonymous

Nope

- anonymous

expand the root term by binomial theorem

- anonymous

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

- anonymous

while opening with binomial neglect higher degree terms

- anonymous

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?

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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 .....

- anonymous

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.

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

We haven't studied about that rule yet

- anonymous

do u want to learn it?

- anonymous

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

- shubhamsrg

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

- anonymous

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

- shubhamsrg

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

- anonymous

Yeah, I don't underst that part

- shubhamsrg

hmm,,heard of taylor series expansion ?

- anonymous

No

- 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..

- shubhamsrg

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

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