Calculus I, Fall 2011 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 02

Assigned Sep 21, 2011.

1. Section 2.5: problems 30, 32, 35, 38, 41.

2. Chap 2: problems 74, 75.

3. How would you define the following limits formally using ϵ and δ? (Need not prove anything, just define them) a.

lim_{x→c^{+}} f(x) = L

b.

lim_{x→c} f(x) = ∞

c.

lim_{x→−∞} f(x) = L

Hint: The formal definition of lim_{x→c} f(x) = L is a translation of

f(x) can be arbitrarily close to L as long as x ≠ c is close enough to c The ‘f(x) → ∞’ part, in plain words can be like ‘f(x) be arbitrarily large’ while the ‘as x → ∞’ part can be ‘whenever x is large enough’. The latter, in mathematical language, would be ‘there is an M such that for all x > M, ⋯’

4. Use the ϵ − δ argument to show that, if f(x) and g(x) are both continuous at x = c, then so is f(x) + g(x) and f(x) − g(x).

5. Use the ϵ−δ argument to show that if lim_{x→c}f(x) = L and lim_{y→L}g(y) = M then lim_{x→c}g(f(x)) = M.

6. Use the ϵ − δ argument to show that if f(x) is continuous at x = c, then so is 3f(x).

7. Section 3.1: problems 35, 40, 49.