Proof of a limit point on a plane.
Prove that every infinite and bounded point collection in the plane (R2) has a limit point.
Working with the limit of Supremum.
Let {En} be a collection of non-empty sets. Show that LimSupEn={x: x is in En for infinitely many n}
If v is a signed measure, E is v-null if |v|(E)=0
Suppose that {x_n} is a sequence which satisfies |x_{n+1} - x_n| <= 1/log n Is this sequence Cauchy? What about the one satisfying |x_{n+1} - x_n| <= 1/(1 + epsilon)^n where epsilon > 0?
Using the definition of a limit (rather than the limit theorems) prove that lim {x -> a+} f(x) exists and find the limit in each of the following cases a) f(x) = x/|x|, a = 0. b) f(x) = x + |x|, a = -1. c) f(x) = (x - 1)/(x^2 - 1), a = 1. In which cases do lim {x -> a-} f(x) and lim {x -> a} f( ...continues
Evaluate the following limits using the epsilon - delta definition and the limit theorems: a) lim {x -> 0} (x^2 + cos x)/(2 - tan x) b) lim {x -> sqrt(pi)} ((pi - x^2)^(1/3))/(x + pi)
Evaluate the following limits using the epsilon - delta definition and the limit theorems. a) lim {x -> 0} sin x sin (1/x^2) b) lim {x -> Infinity} (x^3 + 1)/(x^3 cos(1/x) + x^2 - 1) Please also show how you came up with the answer.
Where is the function f(x) = (q^2 - 1)/q^2 if x = p/q meaning x is a rational in reduced form and f(x) = 1 when x is not a rational continuous in the interval (0,1)? Please also explain how you came up with the answer.
Suppose that f(x) satisfies the functional equation f(x + y) = f(x) + f(y) for all x,y in R (the real numbers). Prove that if f(x) is continuous that f(x) = cx where c is a constant. What can you say about f(x) if it is allowed to be discontinuous?
Prove that if f(x) = x^alpha, where alpha = 1/n for some n in N (the natural numbers), then y = f(x) is differentiable and f'(x) = alpha x^(alpha - 1). Progress I have made so far: I have managed to prove, (x^n)' = n x^(n - 1) for n in N and x in R both from the definition of differentiation involving the limit and ...continues
