Thomas’ Calculus Early Transcendentals 13ed

Homework Assignment for Chapter 02

1. Section 2.2: 33, 41, 63, 77, 81, 87.

Remark: For problem 87, just find the limit. Need not plot it. To find the limit, use the algebraic identity ‘a^{3} − b^{3} = ⋯’ mentioned in lecture 01 video.

2. Section 2.3: problems 35, 43, 49, 53.

3. Use the ε − δ argument to prove the following statement:

If lim_{x→c} f(x) = L and lim_{x→c} g(x) = M, then lim_{x→c} (4f(x) − 2g(x)) = 4L − 2M. Hint: Note that if a < b then −a > −b.

4. Suppose that f(x) is defined on (c − a, c) ∪ (c, c + a) for some a > 0. If f(x) satisfies the following statement, then is it true that lim_{x→c} f(x) = L? Prove it if true, find a counter example if not true.

For any ε > 0 and any δ > 0, there exists a number x ∈ (c − δ, c) ∪ (c, c + δ) such that |f(x) − L| < ε.

5. (Optional) Section 2.3: Problems 57 (see derivations of this definition in Lecture 03).

6. Section 2.4: Problems 26, 34, 42, 48.

7. Chap 2, Additional and Advanced Exercises: Problems 25 (Hint: 1 − cos x = 2 sin^{2} \frac{x}{2}), 26 on page 136.

8. Section 2.5: problems 37 (just find the limit, ignore the continuity question) 64, 67, 68, 77 (Need not graph it).

9. Section 2.6: problems 61, 85, 92, 93, 95, 97.

10. Study definition of the limits in p102, p119, p125 and p131. How would you define the

four limits

lim_{x→±∞} f(x) = ±∞?

Verify the statement

lim_{x→∞} −x^{3} = −∞

using the definition you wrote.