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Chapter 1: Q. 77 (page 136)

Describe the intervals on which each function f is continuous. At each point where f fails to be continuous, use limits to determine the type of discontinuity and any left- or right-continuity.

fx=sinx,ifx<蟺cosx,ifx鈮�蟺

Short Answer

Expert verified

The function is continuous on-鈭�,蟺,蟺,鈭�and only the right limit is continuous at x=蟺.

Step by step solution

01

Step 1. Given information.

Consider the given question,

fx=sinx,ifx<蟺cosx,ifx鈮�蟺

02

Step 2. Calculate the left and right limits.

We need to calculate the left and right limits,

limx鈫�蟺+fx=limx鈫�蟺+sinx=sin蟺=0limx鈫�蟺fx=limx鈫�蟺cosx=cos蟺=-1limx鈫�蟺-fx=limx鈫�蟺-cosx=cos蟺=-1

As the left and right limits both exists but not equal.

Hence, the function is continuous on -鈭�,蟺,蟺,鈭�and only the right limit is continuous atx=蟺.

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Calculate each of the limits:

limx鈫�1xsin-1x2.

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

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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.

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