I want help in a question that i came across
\(\displaystyle\lim_{x \rightarrow 0}(\sum _{ r=0 }^{ n }{ (-1)^{ r }.^{ n }C_{ r } } (\sum _{ k=0 }^{ n-r }{ ^{ n-r }{ C }_{ k }{ 2 }^{ k }{ x }^{ k } } )(x^{ 2 }-x)^r) ^{\frac{1}{x}}\)
If the value of the limit = eλn. Find λ
My work:
⇒x→0lim(r=0∑n(−1)r.nCr(1+2x)n−r(x2−x)r)x1
⇒x→0lim(r=0∑nnCr(1+2x)n−r(−1)r(x2−x)r)x1
⇒x→0lim(r=0∑nnCr(1+2x)n−r(x−x2)r)x1
⇒x→0lim(1+3x−x2)xn
How to calculate this limit?
#Calculus
#BinomialTheorem
#Limits
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Comments
Answer Should be λ=3 . It is 1∞ form.
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change limit as L=enlimx→0xln(1+3x−x2) now you can use expansion as suggested by taylor series.
Else this is 0/0 form so you can use L-hospital rule , if you had studied it yet.
Else There is direct formula for 1∞ form :
L=limx→af(x)g(x)=elimx→a(f(x)−1)g(x)here:limx→af(x)=1&limx→ag(x)=∞
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Oh thanks
Yeah but explain.
You can use the fact that x→0limf(x)g(x)=x→0limeg(x)(f(x)−1)
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Thanks