Prove there is a sequence {xn}, satisfying lim(xn) =0 as n goes to infinity, such that lim f(xn) is not L.?

Question

Suppose a function f(x) is defined on (-1,1) and lim f(x) is not equal to L as x goes to 0.
Prove that there exists a sequence of numbers {xn} satisfying lim xn as n goes to infinity = 0
such that lim f(xn) is not equal to L as n goes to infinity.

Answer 1

Define f(x) = 0 if and only if x = 0, but equals to 1/2 if x is any other value on (-1, 1)



Now, consider the sequence of xn = abs (x^n) for x in (-1, 1). Observe that xn approaches 0 as n >>0, but f(xn) = 1/2, so it's limit = 1/2

Answer 2

Since the limit as x->0 f(x) is different from L, there exists ε_0 >0 such that for every δ > 0 there exists x such that



0 < |x - 0| < δ but |f(x) - L| ≥ ε_0.



Now consider the sequence of delta values δ_n = 1/n. By the above, we can construct a sequence {x_n} in the following way:



For each n, choose x_n such that



0 < |x_n - 0| < δ_n and |f(x_n) - L| ≥ ε_0.



That is, lim(n->∞) f(x_n) ≠ L. It only remains to show that x_n -> 0.



Notice that for any ε > 0, we can choose N such that N ≥ 1/ε. Then for every n ≥ N we have



|x_n - 0| < 1/n ≤ ε.



Hence x_n -> 0. So we've constructed a sequence {x_n} convergent to 0 and such that f(x_n) does not converge to L.

Answer 3

I don't why. I feel it's in most cases on account that Elvis image manner the bad-boy lover that every person desired. Michael never expressed himself as a "bad-boy" and normally stored a easy report for that reason what ever he does is going to be in its own spotlight, which is the motive why more persons pay attention to Michael and the misconception than Elvis and true fact. Additionally as a rule considering that Priscilla Presley on no account made her relationship with Elvis a tremendous situation like she was forced into it, she showed curiosity in him and showed that she adored him. Not like the little boy who claimed Michael had touched him, who made it a gigantic crisis. The biggest cause that Michael is attacked extra is manly for the reason that he hasn't as soon as stated "sure, I rapped him" or "yes, I touched him without permission." There used to be nothing to confess and since of this Michael was perpetually bothered invariably adopted, Elvis made it identified that he and Priscilla have been courting and had been considering every different thus he wasn't troubled as so much about it. An extra key motive this was once this sort of tremendous challenge is of direction racism. Michael Jackson is typically the most identified African-American on the planet and thus individuals need him to screw-up. They want him to be abusive, or to turn out to be in prison or to head "ghetto" on any individual, which is a tactic Obama has additionally realized of (realize how he under no circumstances offers a speedy response to the clicking question and that he continuously stops to believe it over.) Being that Michael has nearly the cleanest report ever he is a foremost goal. Michael is the greatest entertainer, humanitarian and have an effect on of the ultra-modern day which makes him a essential goal for the clicking who will at all times try to stick a label on him. Trust me simply when you consider that he's useless doesn't mean the hunt on MJ filth is off. He'll be attacked for as long as he will be remembered and that is a truth.