You're welcome. I'm glad I could help.

In summary, to find an upper bound M for f(x) = abs ( x+2 / x-8 ) if abs(x-7) < 1/2, we can use the fact that the numerator, |x+2|, can be no larger than 19/2 and the denominator, |x-8|, can be no smaller than 1/2. This means that the fraction, |x+2|/|x-8|, can be no larger than 19, making 19 the upper bound for f(x).
  • #1
lovemake1
149
1

Homework Statement



Find an upper bound M for f(x) = abs ( x+2 / x-8 ) if abs(x-7) < 1/2

Homework Equations


The Attempt at a Solution



i first found set of x values using abs(x-7) < 1/2
which is 13/2 < x < 15/2.

Now, i believe i have to find other set of x values to compare to find upper bound for x.
But how ? please help.
 
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  • #2
lovemake1 said:

Homework Statement



Find an upper bound M for f(x) = abs ( x+2 / x-8 ) if abs(x-7) < 1/2


Homework Equations





The Attempt at a Solution



i first found set of x values using abs(x-7) < 1/2
which is 13/2 < x < 15/2.

Now, i believe i have to find other set of x values to compare to find upper bound for x.
But how ? please help.
You want to find an upper bound for f(x), not x. If 13/2 < x < 15/2, what are the possible values for |x + 2|/|x - 8|? Look at the numerator and denominator separately, and see what intervals you get for |x + 2| and |x - 8|.
 
  • #3
You don't really need to find the x-limits. If -1/2< x- 7< 1/2, then, adding 9 to each part, 17/2< x+ 2< 19/2. Subtracting 1 from each part, -3/2< x- 8< -1/2.

To make a fraction as large as possible, make the numerator as large as possible and the denominator as small as possible.
 
  • #4
HallsofIvy said:
You don't really need to find the x-limits. If -1/2< x- 7< 1/2, then, adding 9 to each part, 17/2< x+ 2< 19/2. Subtracting 1 from each part, -3/2< x- 8< -1/2.

To make a fraction as large as possible, make the numerator as large as possible and the denominator as small as possible.

Does that mean then

A) (x+2)<19/2 and (x-8)<-3/2, thus (x+2)/(x-8)<-19/3

i.e upper bound <-19/3?

or

B) |x+2|<19/2 and |x-8|<1/2 thus |x+2|/|x-8|<19

i.e upper bound <19 ?
 
Last edited:
  • #5
If you think (A) is possible then (i) you did not read what I said (because you are making both numerator and denominator as large as possible) and (ii) you are not using common sense (because it makes no sense to say that a fraction of absolute values has a negative number as upper bound).

I said, "To make a fraction as large as possible, make the numerator as large as possible and the denominator as small as possible."

The numerator, |x+ 2|, can be no larger than 19/2. The denominator, |x- 8|, can be no smaller than 1/2 (-3/2< x- 8< -1/2 so 1/2< |x- 8|< 3/2).

[tex]\frac{|x+2|}{|x- 8|}[/tex] can be no larger than [tex]\frac{19/2}{1/2}= 19[/tex].
 
Last edited by a moderator:
  • #6
HallsofIvy said:
If you think (A) is possible then (i) you did not read what I said (because you are making both numerator and denominator as large as possible) and you are not using common sense (because it makes no sense to say that a fraction of absolute values has a negative number as upper bound).

I said, "To make a fraction as large as possible, make the numerator as large as possible and the denominator as small as possible."

The numerator, |x+ 2|, can be no larger than 19/2. The denominator, |x- 8|, can be no smaller than 1/2 (-3/2< x- 8< -1/2 so 1/2< |x- 8|< 3/2).

[tex]\frac{|x+2|}{|x- 8|}[/tex] can be no larger than [tex]\frac{19/2}{1/2}= 19[/tex].

HallsofIvy thanks for the reply, and appreciate the clarification, things are making sense now :)!
 

1. What is the upper bound of a limit?

The upper bound of a limit refers to the largest number that a function or sequence approaches as the independent variable approaches a certain value. It represents the highest possible value that the function or sequence can reach, and is often denoted by the symbol "L".

2. How is the upper bound of a limit calculated?

The upper bound of a limit can be calculated by evaluating the function or sequence at different values of the independent variable and observing the resulting values. The upper bound is the largest of these values and can also be found by taking the limit of the function or sequence as the independent variable approaches a certain value.

3. What is the significance of the upper bound of a limit in mathematics?

The upper bound of a limit is an important concept in mathematics as it helps to define the behavior of a function or sequence at a specific point. It also allows us to make predictions about the values of the function or sequence as the independent variable gets closer to a certain value.

4. Can the upper bound of a limit be infinite?

Yes, the upper bound of a limit can be infinite. This occurs when the function or sequence has no limit at a certain point and continues to increase without bound as the independent variable approaches that point. In this case, the upper bound would be represented by the symbol "∞".

5. How does the upper bound of a limit differ from the lower bound?

The upper bound of a limit represents the highest possible value that a function or sequence can approach, while the lower bound represents the lowest possible value. In some cases, the upper and lower bounds may be the same, but in others, they can differ, indicating that the function or sequence has a range of values it can approach at a specific point.

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