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Chapter 1: Q. 67 (page 108)

Prove each of the limit statements in Exercises67鈥�72. You

will have to bound 未.

limx鈫�3x2-2x-3=0

Short Answer

Expert verified

The given limitlimx鈫�3x2-2x-3=0 is proved.

Step by step solution

01

Step 1. Given information.

We are given,

limx鈫�3x2-2x-3=0

02

Step 2. Proving the limit

Given 蔚>0, choose 未=min1,蔚5.

Then if 0<|x-3|<未, we have,

x2-2x-3-0=x2-3x+x-3=|x(x-3)+1(x-3)|=|(x-3)(x+1)|=|x-3||x+1|<未|x+1|

03

Step 3. Proving the limit

Now from, 0<|x-3|<未,

role="math" x-3<1x<1+3x<4

Therefore,

<未|x+1|鈮�未(5)鈮�蔚

Hence Proved.

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Most popular questions from this chapter

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

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

limx鈫�-5(3x-2).

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

limx鈫�0x-3.

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�0(3-4x)=3

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

limx鈫�12x+4=6.
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