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Chapter 1: Q. 58 (page 98)

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.
$\underset{x \to \infty}{l i m} \frac{3 x}{x + 1} = 3, ε = 0.1, find s m a l l e s t N > 0$

Short Answer

Expert verified

The required value of $N = 29$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \frac{3 x}{x + 1}$

02

Step 2. Explanation

From the given function, we have, $c = \infty, L = 3$

The limit expression can be written as a formal statement as below,

For all epsilon positive, there is someN positive such that if $x \in (N, \infty)$

Then $\frac{3 x}{x + 1} \in (3 - ε, 3 + ε)$

Now the smallest value of N is given by,

$\frac{3 x}{x + 1} = 3 - 0.1 \frac{3 x}{x + 1} = 2.9 3 x = 2.9 x + 2.9 0.1 x = 2.9 x = 29$

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Most popular questions from this chapter

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of $f (x)$ as $x \to c$ is the value $f (c)$ .

(c) The limit of $f (x)$ as $x \to c$ might exist even if the value of $f (c)$ does not.

(d) The two-sided limit of $f (x)$ as $x \to c$ exists if and only if the left and right limits of $f (x)$ exists as $x \to c$ .

(e) If the graph of $f$ has a vertical asymptote at $x = 5$ , then $\lim_{x \to 5} f (x) = \infty$ .

(f) If $\lim_{x \to 5} f (x) = \infty$ , then the graph of $f$ has a vertical asymptote at $x = 5$ .

(g) If $\lim_{x \to 2} f (x) = \infty$ , then the graph of $f$ has a horizontal asymptote at $x = 2$ .

(h) If $\lim_{x \to \infty} f (x) = 2$ , then the graph of $f$ has a horizontal asymptote at $y = 2$ .

Calculate each of the limits:

$\lim_{h \to 0} \frac{{(3 + h)}^{2} - 3^{2}}{h}$ .

If $f$ is a continuous function, what can you say about $\lim_{x \to 1} f (x) ?$

Use what you know about one-sided limits to prove that a function $f$ is continuous at a point $x = c$ if and only if it is both left and right continuous at $x = c$ .

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| < $ε$ .

$\lim_{x \to 3} (x^{2} - 2 x - 3) = 0; you may assume δ \leq 1$

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