finding possible values of δ that correspond to ε = 0.2......

[samistumbo]

I am doing alright with all the material from this current Calculus chapter, but I am having a very hard time with these two problems (oictures are attached, but I couldn't figure out how to make them bigger in the preview):

#1
For the limit lim (x→3) (x^3-2x+5)=26
illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1.

ε = 0.2 δ= ?
ε = 0.1 δ= ?

The definition they're referring to is, lim (x→a) f(x)= L if for every number ε > 0 there is a number δ> 0 such that
if 0<|x-a|< δ then |f(x)-L|<ε.



#2
Given that lim (x→4) (4x-6)=10, illustrate Definition 2 by finding values of δ that correspont to ε=0.5, ε=0.1, ε=0.05.
ε=0.5 δ ≤ ?
ε=0.1 δ ≤ ?
ε=0.05 δ ≤ ?

 

[mmm4444bot]

I am having a very hard time

This request for help is rather vague; I cannot determine what you've learned thus far or why you are stuck.

I started with substituting given values for L, a, and ε into the limit definition, and went from there to solving a cubic equation (via technology). That solution gave me the value 3 + δ. I'm not sure what your course has taught you.

When I do these types of exercises, I usually think about the graphical interpretation of the definition [i.e., rectangle drawn around point (a,L) with width 2δ and height 2ε]. That visualization helps me with the symbolism, as I work through the absolute-value inequalities.

 

[samistumbo]

This request for help is rather vague; I cannot determine what you've learned thus far or why you are stuck.

I started with substituting given values for L, a, and ε into the limit definition, and went from there to solving a cubic equation (via technology). That solution gave me the value 3 + δ. I'm not sure what your course has taught you.

When I do these types of exercises, I usually think about the graphical interpretation of the definition [i.e., rectangle drawn around point (a,L) with width 2δ and height 2ε]. That visualization helps me with the symbolism, as I work through the absolute-value inequalities.

It's an online Calculus course. I can't find an example of this in the book or in our other required reading, therefore I don't really know how to do it. These were the only two questions in the 2 sections we covered that I didn't know how to do, so I'm proud. But yes, this is something I don't really know how to do nor do I understand it. I really and truly need a good explanation along with a good visual which the book does not provide.

 

[mmm4444bot]

I can't find an example of this in the book or in our other required reading

I really and truly need a good explanation along with a good visual which the book does not provide

Well, if your course is not teaching you the material, then these exercises are probably not that important.

The FreeMathHelp boards are not set up to teach classroom material.

If you would like to investigate lessons and examples elsewhere on the Internet, click HERE for a starting point (Google search results on keywords: epsilon-delta limit definition lessons examples).

Of course, if any specific questions arise in your studies, feel free to come back here and post them.

 

[samistumbo]

Well, if your course is not teaching you the material, then these exercises are probably not that important.

The FreeMathHelp boards are not set up to teach classroom material.

If you would like to investigate lessons and examples elsewhere on the Internet, click HERE for a starting point (Google search results on keywords: epsilon-delta limit definition lessons examples).

Of course, if any specific questions arise in your studies, feel free to come back here and post them.

It's an online course where homework IS worth points. I'm trying to SIMPLY see an example, so I will know how to do it and I can complete my homework. I've watched the videos for the course and read the material. I came here hoping someone would give me an example... or explain it to me because I can't find a tutor anywhere in my area.... I've asked everyone I know and offered to even pay people. Since my class is online my professor is like 2 hours away and completely pointless to me as a personal teacher. Seems like everyone who knows this stuff is too far away, too busy, or hasn't seen it in 10 years and has forgotten how to do it. Don't mean to make excuses but if I wasn't desperate, I wouldn't be on here begging for answers to questions. Thanks for the link though... I will check it out.

 

[mmm4444bot]

You can think of the first exercise as asking something like the following.

If you want to ensure that the value of f(x) is within epsilon units of L, then how close to the value a does x need to be? That distance is delta.

In other words, as x approaches 3 from the left (or, after x passes 3), within what distance of 3 does x need to be, in order for the value of f(x) to lie within 0.1 units above or below 26?

That distance is the delta you're asked to find (first when epsilon is 0.1 units, and again when epsilon is 0.2 units).

Having f(x) fall within 0.1 units of 26 means that it must lie somewhere within y = 25.9 to 26.1

What value of x makes f(x) = 25.9? How far below 3 is that value of x?

What value of x makes f(x) = 26.1? How far above 3 is that value of x?

These two distances away from 3 are the same distance (when rounded to four places, as required by the exercise), and that distance is delta.

Here is a graph of f(x) zoomed in very close to x = 3. The function graph is shown in green, and the limit value is shown in red.

The distances of epsilon and delta (on either side of x = 3 and f(x) = 26, respectively) form a rectangle around the point (3,26).

I hope that this image helps you to understand the meaning of epsilon and delta.

 

[burakaltr]

I am doing alright with all the material from this current Calculus chapter, but I am having a very hard time with these two problems (oictures are attached, but I couldn't figure out how to make them bigger in the preview):

#1
For the limit lim (x→3) (x^3-2x+5)=26
illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1.

ε = 0.2 δ= ?
ε = 0.1 δ= ?

The definition they're referring to is, lim (x→a) f(x)= L if for every number ε > 0 there is a number δ> 0 such that
if 0<|x-a|< δ then |f(x)-L|<ε.



#2
Given that lim (x→4) (4x-6)=10, illustrate Definition 2 by finding values of δ that correspont to ε=0.5, ε=0.1, ε=0.05.
ε=0.5 δ ≤ ?
ε=0.1 δ ≤ ?
ε=0.05 δ ≤ ?

I'll do for the first case and you follow the same pattern

0<abs(x-a)<delta

abs(f(x)-L)<eta

at a=4

abs(x-4)<delta

abs(4x-16)<eta

abs(x-4)<eta/4 which satisfies eta = 4 * delta

when eta=0.5 ; delta=0.5/4 = 0.125