Limit for \(\displaystyle\frac{{\left({1}-{\cos{{x}}}\right)}^{{{\left({k}+{x}\right)}}}}{{x}}\) when \(\displaystyle{x}\to{0}\)

aanvarendbq28 2022-03-23 Answered
Limit for (1cosx)(k+x)x when x0
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Answers (2)

German Ferguson
Answered 2022-03-24 Author has 18 answers
Hint:
(1cosx)(32π+x)x=(1cosx)(32π+x1)(1cosxx)
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Nyla Trujillo
Answered 2022-03-25 Author has 9 answers
Since (1cosx)x1 as x0 (prove this!) the desired limit is equal to the limit of (1cosx)kx as x0. Further note that 1cosxx212 hence the desired limit is equal to the limit of 2kx2k1. Thus the desired limit is equal to 12 if k=1/2 and it is 0 if k>1/2 and diverges if k<1/2. All this is valid for x0+. When we take into account x0 then the limit is 12 for k=1/2.
Thus to conclude, the limit does not exist if k12 and is 0 if k>1/2
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