Question

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

Answer #1

Prove that (n + 1)! < nn whenever n > 3.
Conclude that lim as n approaches infinity of n!/nn
=0

Let f be defined on the (0,infinity). Prove that the limit as x
approaches infinity of F(x) =L if and only if the limit as x
approaches 0 from the right of f(1/x) = L. Does this hold if we
replace L with either infinity or negative infinity?

a. Show that if lim sn = infinity and k > 0, then
lim (ksn) = infinity.
b. Show that lim sn = infinity if and only if lim
(-sn) = -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.

Calculate the limit if n approaches Infinity:
lim [(5cosn+3sinn+lnn^7)/n - 1/(7^n) - (100^n)/n! ]

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

Estimate the lim as f(x) approaches -infinity by graphing
f(x)=sqrt{x^{2}+x+9}+x
(b) Use a table of values of f(x) to guess the
value of the limit. (Round your answer to one decimal place.)
(c) Prove that your guess is correct by evaluating
lim x→−∞ f(x).

7.5.
Prove the following:(a) lim n→∞ an = a = ⇒ lim n→∞ |an|=|a|. (b)
limn→∞an=0 ⇐ ⇒ lim n→∞|an|=0

Prove that a sequence (un such that n>=1)
absolutely converges if the limit as n approaches infinity of
n2un=L>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

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