Real Analysis Answers

Let the function f(x) be continuous at a point X_o. Assume that f(X_o) > 2. Prove that there exists a neighborhood of X_o such that f(x) > 2 for every x in this neighborhood.

Prove (using the "epsilon - delta" definition) that the function f(x) = 4x^2 - 5x + 3 is continuous at every point X_0.

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