Real Analysis Answers

Using the definition of the limit at infinity, verify that

0 does not equal the limit x-->infinity cosx.

let x be a nonempty set and let f: X->R have bounded range in R. if a element R , show that
sup(a+f(x): x element X)= a + sup (f(x): x element X)
and
inf(a+f(x): x element X)= a + inf (f(x): x element X)

if a,b in R show that |a+b|=|a|+|b|

Using the definition of the limit at infinity verify that:

lim x-->infinity cos^2(x)/2x^2 = 0.

Using the definition of the limit at infinity verify that
0 does not equal lim x-->infinity cosx.

Using the definition of the limit at infinity verify that
0 does not equal lim x-->infinity cosx.

Using the epsilon-delta definition of the limit prove that if lim x-->a f(x) and lim x-->a g(x) exist, then lim x-->a [f(x)+g(x)] = lim x-->a f(x) + lim x-->a g(x).

Using the definition of the limit at infinity verify that

lim x-->infinity cos^2(x)/2x^2 = 0

Using the epsilon-delta definition verify that
lim x-->3 (x^2-x) = 6.

Using the epsilon-delta definition of the limit, prove that if lim x-->a f(x) and lim x-->a g(x) exist, then
lim x-->a [f(x) + g(x)] = lim x-->a f(x) + lim x-->a g(x).