Calculus I, Fall 2015

Brief answers to Quiz 1

Oct 06, 2015

1. Give formal definition of lim_{x→c} f(x) ≠ L (just the definition, need not find ϵ or δ, etc.). Ans: See Remark 2 in ‘Study Guide for Chap 02 (New!)’.

2. Find lim_{θ→0} \frac{sin(1 − cos θ)}{tan^{2} θ}.

Ans: 1/2. (See the hint in problem 3 of Homework 02)

3. State the Intermediate Value Theorem (Need not prove). Use it to show that x^{x} = 2 has a root. Ans: For the second part, apply Intermediate Value Theorem to the function g(x) = x^{x} − 2 on the interval [1, 2].

4. Give formal definition of ‘y = f(x) is continuous at x = c’ in terms of ϵ and δ. Then use the ϵ − δ argument to show that if both f(x) and g(x) are continuous at x = c, then so is f(x) + g(x). Ans: The first part: For any ε > 0, there exists δ > 0 such that,

|x − c| < δ ⟹ |f(x) − f(c)| < ε

The second part: The proof is similar (and almost identical) to Example 6 of section 2.3. Just replace L and M by f(c) and g(c), respectively. See page 81 for details.

5. Give formal definitions of the following limits (Just the definition, need not find δ).

(a) lim_{x→c^{−}} f(x) = L (b) lim_{x→−∞} f(x) = ∞

Ans:

(a) For any ε > 0, there exists δ > 0 such that, c − δ < x < c ⟹ |f(x) − L| < ε

(b) For any B > 0, there exists N < 0 such that x < N ⟹ f(x) > B