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dglennuo

dglennuo

Answered question

2022-05-28

lim x ( 4 5 + cos x ) prove that the limit is not finite while x using Cauchy definition

Answer & Explanation

stacan6t

stacan6t

Beginner2022-05-29Added 13 answers

Suppose that limit L exists. For ϵ = 1 10 such that x > M 1 | f ( x ) L | < 1 10
Let N 1 N be such that 2 π N 1 > M 1 . Then if
n > N 1 2 n π > 2 π N 1 > M 1 | f ( 2 n π ) L | < 1 10 | 2 3 L | = | 4 5 + cos ( 2 n π ) L | < 1 10 Next for
ϵ = 1 15 , M 2 such that x > M 2 | f ( x ) L | < 1 15 . Again, let N 2 N be such that ( 2 N 2 + 1 ) π > M 2 . So if
n > N 2 ( 2 n + 1 ) π > ( 2 N 2 + 1 ) π > M 2 | f ( ( 2 n + 1 ) π ) L | < 1 15 | 1 L | < 1 15
Thus if n > N 1 + N 2 1 3 | 1 L | + | 2 3 L | < 1 10 + 1 15 = 1 6 1 3 < 1 6
Contradiction. So the limit Lcannot exists

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