# Limits with inequalities

• Oct 9th 2011, 12:22 PM
Barthayn
Limits with inequalities
Hi, I am just wondering if I have the correct understanding of the questions below (in image)

http://img502.imageshack.us/img502/9...lemset3.th.gif

For question 1. I got the highest value of the function is 19. I got this by subbing in 7.5 because the |x - 7| < 0.5 because it is looking 0.5 units within x=7. So the highest value has to be when it is either 6.5 or 7.5. It turns out f(7.5) = 19 was the highest value. Therefore, the answer is 19 correct? I am wondering because it does not include 7.5, therefore the highest value of the function has to be the limit of f(x) as x approaches 7.5, which is 19.

For question two. The 0.3 units of 22. The 22 is the y-value of the function correct?
• Oct 9th 2011, 01:01 PM
Plato
Re: Limits with inequalities
Quote:

Originally Posted by Barthayn
Hi, I am just wondering if I have the correct understanding of the questions below (in image)
For question 1. I got the highest value of the function is 19. I got this by subbing in 7.5 because the |x - 7| < 0.5 because it is looking 0.5 units within x=7. So the highest value has to be when it is either 6.5 or 7.5. It turns out f(7.5) = 19 was the highest value. Therefore, the answer is 19 correct?

If $|x-7|<0.5$ that means $x\in(6.5,7.5).$
Now on an open interval $\left| {\frac{{x + 2}}{{x - 8}}} \right|$ cannot have a maximum.
BUT you are right 19 is a upper bound.
• Oct 9th 2011, 01:18 PM
Barthayn
Re: Limits with inequalities
Just as I was thinking over a few minutes ago. So the lower bound is 17/3 while the upper bound is 19. The highest value between these is no value at all because you can always get closer to 16 while not touching 16. Thanks for aid.

With question 2 though. The y-value (the limit) is 22 correct?
• Oct 9th 2011, 02:24 PM
Plato
Re: Limits with inequalities
Quote:

Originally Posted by Barthayn
With question 2 though. The y-value (the limit) is 22 correct?

For the second one we want
$|(5x+2)-22|=|5x-20|<0.3$.
Factor out the 5 to get $|x-4|<0.06.$