Question

a. Show that if lim s_{n} = infinity and k > 0, then
lim (ks_{n}) = infinity.

b. Show that lim s_{n} = infinity if and only if lim
(-s_{n}) = -infinity

Answer #1

(1) Show that -infinity to infinity ∫ xdx is divergent.
(2) Show that lim t→∞ definite integral -t to t ∫ xdx = 0. This
shows that we cannot define ∫ ∞ −∞ xdx= lim t→∞ definite integral-1
to t ∫ xdx

Prove that lim n^k*x^n=0 as n approaches +infinity. Where
-1<x<1 and k is in N.

using theorem "the sequential characterization of limit" show
lim(c/x^2) =0 as x goes to infinity for all c in R.

Let {s_n} be a sequence of positive numbers. Show that the
condition lim as n-> infinity of (s_n+1)/(s_n) < 1 implies
s_n -> 0

If lim Xn as n->infinity = L and lim Yn as n->infinity =
M, and L<M then there exists N in naturals such that Xn<Yn
for all n>=N

Show that the series sum(an) from n=1 to infinity
where each an >= 0 converges if and only if for every
epsilon>0 there is an integer N such that | sum(ak )
from k=N to infinity | < epsilon

1. if lim
x->infinity+ (5+2X)/(9-6X)=??
2. what is the value for the same equation if lim x approaches
to negative infinity???
3. if lim x approaches to positive infinity then evaluate the
equation (4X+7)/(5X^2-3X+10)
4. What would be the value of the above mentioned equation is
lim x approaches to negative infinity

Prove that if (xn) is a sequence of real numbers,
then lim sup|xn| = 0 as n approaches infinity. if and
only if the limit of (xN) exists and xn
approaches 0.

prove if lim?→∞ an = a>0 and if lim?→∞ sup bn = b (bn≥0) then
lim?→∞ sup anbn =ab
0<R<∞ : an∈R

What are the horizontal and vertical asymptotes of
1/(1+e^x).
And compute the lim x infinity and negative infinity to evaluate
the end behaviour.

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