   Chapter 2.2, Problem 55E

Chapter
Section
Textbook Problem

(a) Use numerical and graphical evidence to guess the value of the limit lim x → 1 x 3 − 1 x − 1 (b) How close to I does x have to be to ensure that the function in pan (a) is within a distance 0.5 of its limit?

(a)

To determine

To guess: The value of limx1x31x1 with the numerical and the graphing evidence.

Explanation

Calculation:

As x approaches 1, evaluate the function f(x)=x31x1 for the numbers 0.99,0.999,0.9999,1.01,1.001 and 1.0001.

Evaluate the function (correct to 6 decimal places) for the values of x and get the following table of values.

 x x3−1 x−1 f(x)=x3−1x−1 0.99 −0.0297 −0.005012563 5.925312 0.999 −0.003 −0.000500125 5.992503 0.9999 −0.0003 −0.0000500013 5.99925 1.01 0.030301 0.004987562 6.075313 1.001 0.003003 0.000499875 6.007503 1.0001 0.0003 0.0000499988 6.00075

From table, it is observed that the value of f(x)=x31x1 gets closer to 6 as x approaches to 1 from the either side

(b)

To determine

How close to 1 does 5.5x6.5 in part (a).

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