Calculus I, Fall 2015 (http://www.math.nthu.edu.tw/˜ wangwc/)

Brief answer to selected problems in HW02

1. No. For example. lim_{x→1} f(x) = 1. Yet, for any δ > 0, one can take ϵ_{0} = δ/4 and x_{0} = 1 + δ/4 satisfying the definition.

2. Section 2.4: Problem 26 (answer = 2), 34 (answer = 1), 42 (answer = 1).

3. Section 2.5:

Problem 64: If f is discontinuous at g(0), f(g(x)) may still be discontinuous at x = 0. No contradiction.

4. Chapter 2: Problem 25 (answer = 0), 26 (answer = 0).

5. Section 2.5:

Problem 67: The function g(x) = f(x) − x satisfies g(0) · g(1) ≤ 0. If the equality holds, then c is found. Otherwise, apply the Intermediate Value Theorem to g(x) on [0, 1]. Problem 77: The function f(x) = cos(x) − x satisfies f(0) · f(1) < 0.

6. Section 2.6:

Problem 92: If B > 0, take δ = 1/\sqrt{B} will do. If B ≤ 0, take δ to be any positive real number (for example δ = 1) will do. Problem 93: (a): change ’x_{0} < x < x_{0} + δ’ to ’x_{0} − δ < x < x_{0}’. (b): change ’f(x) > B to f(x) < B (or f(x) < −B). (c): apply both changes in (a) and (b). Problem 100: y = x + 1 + \frac{2}{x−1}. Vertical asymptote: x = 1. Oblique asymptote: y = x + 1.