Finding the largest value of delta greater than 0

In summary: At any rate:\displaystyle |\sqrt{27-x\ } - 4|+4\ne |\sqrt{27-x\ }|if \displaystyle |\sqrt{27-x\ } - 4| < 1\ .
  • #1
zurburgk
2
0

Homework Statement


For f(x) = sqrt(27-x), L=4, x_0 = 11 epsilon = 1, find the largest value of delta > 0 in the formal definition of a limit which ensures that |f(x) - L| < epsilon


Homework Equations



the formal def. of a limit:
lim x->x_0 F(x) = L if, for every number epsilon > 0, there exists a corresponding number delta>0 such that
0<|x-x_0| < delta implies |f(x) - L| < epsilon


The Attempt at a Solution



I've tried plugging multiple different things into the equations, but i, sadly, have no idea what I'm doing.

0 < |x-11| < epsilon and |sqrt(27-x) - 4| < 1.

Attempting to solve the second equation i get x>2. If i plug that into the first equation i got -9. But that is definitely not the answer. I just don't know what all these numbers mean.
 
Physics news on Phys.org
  • #2
zurburgk said:

Homework Statement


For f(x) = sqrt(27-x), L=4, x_0 = 11 epsilon = 1, find the largest value of delta > 0 in the formal definition of a limit which ensures that |f(x) - L| < epsilon

Homework Equations



the formal def. of a limit:
lim x->x_0 F(x) = L if, for every number epsilon > 0, there exists a corresponding number delta>0 such that
0<|x-x_0| < delta implies |f(x) - L| < epsilon

The Attempt at a Solution



I've tried plugging multiple different things into the equations, but i, sadly, have no idea what I'm doing.

0 < |x-11| < epsilon and |sqrt(27-x) - 4| < 1.

Attempting to solve the second equation i get x>2. If i plug that into the first equation i got -9. But that is definitely not the answer. I just don't know what all these numbers mean.
How about solving [itex]\displaystyle |\sqrt{27-x\ } - 4| = 1[/itex] for x, to start things off ?
 
  • #3
Do you mind if i walk you through my steps, if i make a mistake it should be easier to pinpoint the area :)

So |sqrt(27-x)-4| < 1
- add 4 to both sides

|sqrt(27-x)| < 5
-get rid of the square root

|27-x|< 25
-subtract 27 from each side

|-x| < -2

- and here is where I'm having my first problem. Do i multiply both sides by a negative allowing x to be a positive variable? Or do the absolute value signs take care of the negative variable?
 
  • #4
zurburgk said:
Do you mind if i walk you through my steps, if i make a mistake it should be easier to pinpoint the area :)

So |sqrt(27-x)-4| < 1
- add 4 to both sides

|sqrt(27-x)| < 5
-get rid of the square root

|27-x|< 25
-subtract 27 from each side

|-x| < -2

- and here is where I'm having my first problem. Do i multiply both sides by a negative allowing x to be a positive variable? Or do the absolute value signs take care of the negative variable?
Let me repeat,
Solve [itex]\displaystyle |\sqrt{27-x\ } - 4| = 1[/itex] for x.​

But if you insist, you can solve the inequality. Just be careful. That is trickier.

At any rate:
[itex]\displaystyle |\sqrt{27-x\ } - 4|+4\ne |\sqrt{27-x\ }|[/itex]​

If [itex]\displaystyle |\sqrt{27-x\ } - 4| < 1[/itex]

then [itex]\displaystyle -1<\sqrt{27-x\ } - 4 < 1\ .[/itex]

Can you continue from there?
 

What is the significance of finding the largest value of delta greater than 0?

Finding the largest value of delta greater than 0 is important in many mathematical and scientific calculations. It represents the maximum allowable change in a variable or quantity without causing significant disruption to the system or experiment.

How is the largest value of delta greater than 0 calculated?

The largest value of delta greater than 0 is typically calculated using mathematical methods such as derivatives, optimization algorithms, or trial and error. It can also be determined through experimental data and statistical analysis.

What factors can affect the largest value of delta greater than 0?

The largest value of delta greater than 0 can be influenced by a variety of factors such as the accuracy of measurements, the complexity of the system or experiment, and the desired level of precision. It may also be impacted by external factors such as environmental conditions or equipment limitations.

Why is it important to have a value of delta greater than 0?

A value of delta greater than 0 ensures that there is room for change and variability in a system or experiment. This allows for more accurate and reliable results, as well as the ability to account for unexpected factors or errors.

Can the largest value of delta greater than 0 change over time?

Yes, the largest value of delta greater than 0 can change over time as the system or experiment evolves or as new data is collected. It is important to continually reassess and adjust this value to ensure the accuracy and validity of calculations and conclusions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
870
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
26
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
983
  • Calculus and Beyond Homework Help
Replies
20
Views
1K
Replies
1
Views
620
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Back
Top