For each of the following prove or disprove the limit

Question

Assignment 2-

1. Let n ≥ 1 be a positive integer. Prove

1³ + 2³ + · · · + n³ = n²(n + 1)²/4.

2. Prove that for any integer n ≥ 1,

(cos(θ) + I sin(θ))ⁿ = cos(nθ) + I sin(nθ),

where i² = -1.

3. For each of the following, prove or disprove the limit exists.

(a) lim_x→3(2x - 1).

(b) lim_x→3 x².

(c) lim_x→0 1/x.

4. Give examples to show that the following definitions of lim_x→a f(x) = l are not correct.

(a) For all δ > 0 there is an ∈ > 0 such that if 0 < |x - a| < δ then |f(x) - l| < ∈.

(b) For all ∈ > 0 there is a δ > 0 such that if |f(x) - l| < ∈ then 0 < |x - a| < δ.

5. Define the function f(x) on the interval [0, 1] by

Determine, with proof, whether or not

limx→_9/2012f(x) = f(9/2012).

Also, determine, with proof, whether or not

lim_x→1/√2f(x) = f(1/√2).

6. Compute the following limits (you don't need to use the definition of a limit at all):

(a) lim_x→∞(√(x+1) - √x).

(b) lim_x→0 tan(3x)/x.

(c) lim_x→∞(7^x + 2^x)^1/x.