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Chapter 1: Q. 5 (page 152)

what we mean when we say that a limit exists, or that a limit does not exist.

Short Answer

Expert verified

If a limit exists then either x comes from a positive direction or a negative direction, function value approaches the same value.

If a limit does not exist then either the graph of the function has vertical asymptotes or the function has a gap at any value of x.

Step by step solution

01

Step 1. Condition of a limit exists.

If a limit exists then either x comes from a positive direction or a negative direction, function value approaches the same value.

02

Step 2. Condition for a limit does not exist.

If the graph of the function is discontinuous or has any hole or gap at x value cthen the limit does not exist.

If the graph of the function has a vertical asymptote at x value cthen the limit does not exist.

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Most popular questions from this chapter

For each functionf graphed in Exercises23鈥�26, describe the intervals on whichf is continuous. For each discontinuity off, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

For each limit statement , use algebra to find 未 > 0 in terms of 蔚> 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < 蔚.

limx鈫�12x4=2;youmayassume未鈮�1.

Calculate each of the limits:

limx鈫�蟺2cotxcosx.

Write each of the inequalities in interval notation:

fx-L<鈭�

State what it means for a function f to be left continuous at a point x = c, in terms of the delta鈥揺psilon definition of limit.
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