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Chapter 1: Q. 3 (page 97)

What are punctured intervals, and why do we need to use them when discussing limits?

Short Answer

Expert verified

The punctured interval is defined on $(c - δ, c) \cup (c, c + δ)$ where $δ > 0$ represents a small distance to the left and right of $x = c$ .

Step by step solution

01

Step 1. Given information.

What are punctured intervals, and why do we need to use them when discussing limits?

02

Step 2. Statement.

The punctured interval is defined on $(c - δ, c) \cup (c, c + δ)$ , where $δ > 0$ represents a small distance to the left and right of $x = c$ .

The limits are never be concerned with what happens at the point $x = c$ , only near the point $x = c$ .

If we are calculation the value at $x = 3$ for the function $f (x) = 4 x$ , then we can use both $f (2.99) and f (3.01)$ within any small unit of $f (3) = 4 (3) = 12$ .

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Most popular questions from this chapter

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x \to 1} (2 x^{2} - 4 x + 3) = 1$ .

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\lim_{x \to 0} x^{- 3} .$

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

$\lim_{x \to 0^{-}} f (x) = - 1, \lim_{x \to 0^{+}} f (x) = 1, f (0) = 0 .$

For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .

$\lim_{x \to 1} 2 x^{4} = 2; you may assume δ \leq 1$ .

$\lim_{x \to + 2} \sqrt{2^{x} - 4}$

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