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Chapter 1: Q. 80 (page 136)

Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.

limx鈫�0xsin1x2

Short Answer

Expert verified

The limit of the given equation is 0.

Step by step solution

01

Step 1. Given information.

Consider the given question,

limx鈫�0xsin1x2

02

Step 2. Apply squeeze theorem.

Range of sin1x2or sinx2is -1,1.

localid="1648747938243" -1鈮�sin1x2鈮�1

Multiply x on all the sides,

localid="1648747930903" -1x鈮�xsin1x2鈮�1x-x鈮�xsin1x2鈮�x

Applying limits on all the sides as x鈫�0,

localid="1648747951469" limx鈫�0-x鈮�limx鈫�0xsin1x2鈮�limx鈫�0x-0鈮�limx鈫�0x2sin1x2鈮�0

Applying the Squeeze theorem,

=0

03

Step 3. Plot the graph.

Representing the graph, we get,

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Most popular questions from this chapter

Sketch the graph of a function f described in Exercises 27鈥�32, if possible. If it is not possible, explain why not.

f is left continuous at x = 1 and right continuous at x = 1, but is not continuous at x = 1, and f(1) = 鈭�2.

limx鈫�0sinxcosx

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�1(3x+8)=11

Calculate each of the limits:

limh鈫�0(1+h)3-13h.

For each limit limx鈫�cf(x)=Lin Exercises 43鈥�54, use graphs and algebra to approximate the largest value of 未such that if localid="1648023101818" x鈭�(c-未,c)鈭�(c,c+未)thenf(x)鈭�(L-蔚,L+蔚)

localid="1648023199049" role="math" limx鈫�-1x2-2x-3x+1=-4,蔚=1
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