Suppose that xn is a sequence of real numbers that converges to 1 as n .

Question:

Suppose that xn is a sequence of real numbers that converges to 1 as n → ∞. Using Definition 2.1, prove that each of the following limits exists.
a) 1 + 2xn → 3 as n → ∞.
b) (πxn - 2)/xn → π - 2 as n → ∞.
c) (x2n - e)/xn → 1 - e as n → ∞.