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Chapter 1: Q. 42 (page 98)

Write each limit in Exercises 19鈥�42 as a formal statement involving未,蔚,N,and/orM, and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

limx鈫�鈭�fx=-鈭�

Short Answer

Expert verified

For all M<0, there exists some N>0such that if,

x鈭�N,鈭�thenf(x)鈭�-鈭�,M.

The graph is

Step by step solution

01

Step 1. Given information.

The given expression islimx鈫�鈭�fx=-鈭�.

02

Step 2. Explanation. 

From the given expression, we have, c=鈭�,L=-鈭�.

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

For all M<0, there exists some N>0such that if,

x鈭�N,鈭�thenf(x)鈭�-鈭�,M.

03

Step 3. Graph the expression. 

The graph illustrating the roles of the constants is shown as below:

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Most popular questions from this chapter

Write each of the inequalities in interval notation:

fx-L<鈭�

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Extreme Value Theorem, and illustrate on your graph that the conclusion of the Extreme Value Theorem does not necessarily hold.

limx鈫�蟺1cosec(蟺-x)

In Exercises 39鈥�44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

f(x)=(x-3),ifx<3-(x-3),ifx猢�3.

Write delta-epsilon proofs for each of the limit statements limx鈫�cfx=Lin Exercises 47-60.

limx鈫�-31-x=4.
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