2 LIMITS

CHAPTER 2SAMPLE EXAM(c) For all x with −1 ≤ x ≤ 1, −1 ≤ f (x) ≤ 1(d) Given any y in [−1, 1], then y = f (x) for some x in [−1, 1].(e) If x < −1orx > 1, then f (x) < −1or f (x) > 1.(f) f (x) =−1forx < 0and f (x) = 1forx > 0.{2x2if x ≥−16. Consider the function f (x) =x + 2 if x < −1(a) Let L = lim f (x). FindL.x→0(b) Find a number δ > 0sothatif0< |x| < δ, then | f (x) − L| < 0.01.(c) Show that f does not have a limit at −1.(d) Explain what would go wrong if you tried to show that lim f (x) = 1usingtheε-δ definition.x→−1Hint: Try ε = 1 2 . 81