The question is about proving limits

Question

a) Show that if x_n≠0, where n(∈) is a subset of N then lim(x_n)=0 if and only if lim(1/x_n²)=infinity 

 

b) Give an example for which lim(x_n)=0 but lim(1/x²_n) does not exist

 

Answer 1

(A) ------------>
given: lim(Xn)=0
prove: lim(1/Xn) = infinity

lim (1/Xn) = lim 1 / lim (Xn)

= 1 / limXn

Since lim Xn=0, 1/ lim Xn is infinity

Therefore lim 1/Xn^2 = lim (1/Xn)^2 is also infinity


<-------------------

given : lim 1 / (Xn^2) = infinity
prove: lim Xn=0

By contradiction, suppose lim Xn <= k < infinity is bounded above.

lim 1/Xn <= 1/k < infinity

lim 1/Xn * 1/Xn <= 1/k * 1/k = 1/k^2 < infinity

lim 1/Xn^2 <= 1/k^2 < infinity, which is a contradiction


(b) 1/e^n tends to zero

But the reciprocal blows up