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Valdas
Finding a limit
\[\lim_{x \rightarrow 0} \frac{ \sqrt[5]{1+2x}-1 }{ \sin x }\]
sorry, I forgot to mention that we haven't learnt about l`hopital
you learned Taylors series?
expand the root term by binomial theorem
then there will be term like 10x/sinx and x/sinx=1 for given condition so ans =10
while opening with binomial neglect higher degree terms
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?
no see it i have shown|dw:1353844296253:dw|
and i have done mistake while doing this it is 2/5=0.4 so sorry for that
Could you explain how did you get \[\sqrt[5]{1+2n}=1+\frac{ 2 }{ 5 }n + ....?\]
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 .....
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.
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
Perhaps anyone can think of other way to find the limit? Every problem until this one required some sort of quite simple algebraic manipulation.
actually it is one of the shortest methods but indeed u can initiate in ur problem by factorizing the numerator term
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
I can't use l'hoptial's rule to find the limit
i do not get you.Do you mean ,you are not allowed to use l hopitals rule or you can not evaluate.
We haven't studied about that rule yet
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
@RajshikharGupta 's method seems to be the shortest to me,
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
are you asking how do we get (1+x)^n = 1+xn (given |x|<1 ) ?
Yeah, I don't underst that part
hmm,,heard of taylor series expansion ?
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..
make that substitution in your question.. you'll reach the ans directly then..