E. i. Prove directly from the definitions that if g : R → [1, ∞) is a function so that limx→ c g(x) = L, then lim, ii. Prove the same result of the previous part, using Relating Sequences to Functions. C 1/g(x) = 1/L.

9. For a function f(x) to be continuous at a point x = c, the following three conditions must hold: i. lim f(x) exists ii. f(c) exists iii. lim f(x) = f(c) The following function has discontinuities at x = -1, x = 1, and x = 2. 4. 3- 2- -1 -2 -3 -4 a. For which of the three discontinuities (if any, maybe more than one) is the discontinuity because i (above) fails to hold? b. For which of the three discontinuities (if any, maybe more than one) is the discontinuity because ii (above) fails to hold? c. Which of the three discontinuities (if any, maybe more than one) are removable discontinuities that could be removed by redefining the function at the point of discontinuity?

1. Prove whether lim f(x) exists or does not exist for x→0 (а) f(х) — [1 if x = 0 1 (b) f(x) = if x 20 if x + 0. if x < 0.

where a nл. D - X (v)ɔsɔ — (x)ɔsɔ D+X lim