Expert: Paul Klarreich - 9/17/2007

Question
I am confused on what this problem is asking. It says make a guess at the
limit (if it exists) by evaluating  the function at the specified x-values.

lim x-->1     x - 1 / x^3 -1

x = 2, 1.5, 1.1, 1.01, 1.001, 0. 0.5, 0.9, 0.99, 0.999

How do I evaluate the function. Thank you.

Answer
Questioner:   Jason
Category:  Calculus
Private:  No
 
Subject:  Limits - calc 1
Question:  I am confused on what this problem is asking. It says make a guess at the limit (if it exists) by evaluating  the function at the specified x-values.

lim x-->1     x - 1 / x^3 -1

x = 2, 1.5, 1.1, 1.01, 1.001, 0. 0.5, 0.9, 0.99, 0.999

How do I evaluate the function. Thank you.
.........................................
Hi, Jason,

For the first part, making a guess at the limit, the sensible way is to use your calculator or an Excel spreadsheet (very good for this kind of stuff).  Be sure, of course, to parenthesize correctly, like

=(A1-1)/(A1^3-1)

Then you just put in your values and you should find that the values start getting very close to 0.3333..., so you make that your guess.

Then:  You cannot evaluate the function at  x = 1, because  1^3 - 1 = 0, so you get 0/0, which is indeterminate.

However, since  x = 1 makes the denominator zero,  x-1  must be a factor of the denominator.  (You learned that in high school.)  So you can use that fact to determine the limit (I am sure that is what the problem says to do, anyway.)
      x - 1
lim   -------- =
x->1  x^3 - 1

            x - 1
lim   --------------------- =
x->1  (x - 1)(x^2 + x + 1)

          1
lim   -----------
x->1  x^2 + x + 1

NOW you can try your  x = 1
     1        1
= --------- = ---
 1 + 1 + 1    3