MAT137Y5 Lecture 7: Lecture 7

[3] 7. Give an e-8 proof that lim (1 – 3.r) = 4. Solution: Pick any e > 0 and let := If +11 <0, then \f(x) - L = |(1 – 3x) - 4 = 1 - 3x - 31 = 1 - 3||* +11

10. Give an e-d proof that lim(-2x + 1) = -3. Solution: Pick any e > 0 and let 8=. If |x - 21 < 8, then f (x) - L = 1 - 2.0 +1-(-3) = 1 - 2.0 + 4 = -2|- 21

4. An erroneous proof of a statement is given below. Indicate where a fundamental error in the argument is made. Briefly explain what the error is. The statement is false but this is not relevant to your answer. The lines of the proof have been labelled to make it easier for you to answer. [2 marks] Statement: Let a and b be positive real numbers. If b > 5, then log, (250) <3. Proof: (1) Let a, b e R where a, b > 0. (2) Assume b>5. (3) Using rules of logarithms, log(250) = logo 25+ logo a. (4) Since b > 5, then log, 25 = logo 52 <2. (5) When a = b, log, a = 1. (6) Therefore logo (25a) < 2+1 = 3.