Limit[x->2] [cos(pi/x) / (x-2)], using "hint" t = pi/2 - pi/x, so pi/x = pi/2 - t, so.... ??

[bbm25]

I do not know where they got the 'hint' from, and consequently cannot understand the following working that has also been circled. Please help me to understand this?

 

[bbm25]

What i do understand:
1) the limit has to be substituted with x
2) 0/0 cannot be an answer
3) (sin ax)/bx = a/b

i do not understand:
1) where the 't' came from
2) how to simplify (cos ax)/bx

 

[Dr.Peterson]

You don't really need to know how someone invented a hint in order to use it; but it can help, I suppose.

As I see it, they recognize that the behavior of the cosine near x=2 (that is, when its argument is near pi/2) is essentially a sine, and they want to use the known limit of sin(x)/x. Or, they may just want to shift the function so that the new variable is approaching 0. Either way, they define the new variable as the difference between the argument and pi/2.

(Typically, we see a useful substitution by looking ahead and seeing what might be done with it, so you can discover bits of how they might be thinking as you read through the work. Don't try to understand why they did something without first following what they did, as the expectation of those steps is probably the reason for doing it!)

The work is just a substitution, and it looks like they explain some of the pieces on the right. (I do wish they used more words.) They replace pi/x with pi/2 - t, and (x-2) with 2xt/pi (which is obtained on the right by solving); then they use the cofunction identity to rewrite the cosine as a sine.

If you still have questions, please be as specific as you can.

 

[bbm25]

 

[Dr.Peterson]

Thanks for the details. It would have been kinder of you to type them into your message, so I wouldn't have to retype them below ...

I do not know where t fits in the original expression.

It isn't in the original expression at all. But they saw value in this substitution, which as I said becomes clear as you follow the work.

As I said, they defined t as the difference between the argument of the cosine, pi/x, and the number it approaches, pi/2, as x approaches 0. This allows them to turn the limit into a limit as a variable (t) approaches 0.

I do not know how exactly t approaches 0 as x approaches 2

Isn't it clear that as x approaches 2, pi/2 - pi/x approaches pi/2 - pi/2, which is 0? That's the simplest kind of limit there is.

I do not understand how (x-2) turns into (2xt/pi)

They show you this in the right-hand part of the image you sent!

Given that t = pi/2 - pi/x, you want to solve for x. They did this by combining the fractions on the right using a common denominator, giving t = (pi x - 2pi)/(2x) and factoring out pi to get t = pi(x - 2)/(2x). Then they multiply both sides by 2x/pi to get 2xt/pi = x-2. You could finish by adding 2 to get x = 2xt/pi + 2, but they saw that they only needed an expression for x-2, so they stopped short and just replaced x-2 with 2xt/pi.

I do have to admit that what they did here is a little odd, not having technically solved for x, since they left both x and t in the expression. I imagine they did this because their main interest was to get rid of x-2, and they didn't mind leaving x there because they knew they could handle it. To actually solve for x, the normal thing to do would be to isolate the term with x, pi/x = pi/2 - t = pi/2 - 2t/2 = (pi - 2t)/2, and then take the reciprocal, x/pi = 2/(pi - 2t), then multiply by pi to get x = 2 pi/(pi - 2t). I imagine they saw that this would be unpleasant to work with (and maybe even did that at first and then changed their minds). It would have been nice if they explained their thinking in words, as I've mentioned. It would be more instructive that way.

I also do not understand how to 'cut' (cos ax)/bx into a/b

I don't know what you mean by this. Nothing is cut, and there is no a/b.

What they did is to use the "cofunction identity", cos(pi/2 - u) = sin(u), to rewrite the expression using sin(t).

They also change a division by 2xt/pi into multiplication by pi/(2xt), and then split the limit of a product into a product of limits.

 

[Steven G]

What i do understand:
1) the limit has to be substituted with x
2) 0/0 cannot be an answer
3) (sin ax)/bx = a/b

i do not understand:
1) where the 't' came from
2) how to simplify (cos ax)/bx

1) the limit has to be substituted with x. Not true. At the end you sub 2 for x
2) 0/0 cannot be an answer. Correct
3) (sin ax)/bx = a/b. NOT true at all. What is true is that the lim as x--> 0 of (sin ax)/bx = a/b

 

[bbm25]

Thanks for the details. It would have been kinder of you to type them into your message, so I wouldn't have to retype them below ...

It isn't in the original expression at all. But they saw value in this substitution, which as I said becomes clear as you follow the work.

As I said, they defined t as the difference between the argument of the cosine, pi/x, and the number it approaches, pi/2, as x approaches 0. This allows them to turn the limit into a limit as a variable (t) approaches 0.

Isn't it clear that as x approaches 2, pi/2 - pi/x approaches pi/2 - pi/2, which is 0? That's the simplest kind of limit there is.

They show you this in the right-hand part of the image you sent!

Given that t = pi/2 - pi/x, you want to solve for x. They did this by combining the fractions on the right using a common denominator, giving t = (pi x - 2pi)/(2x) and factoring out pi to get t = pi(x - 2)/(2x). Then they multiply both sides by 2x/pi to get 2xt/pi = x-2. You could finish by adding 2 to get x = 2xt/pi + 2, but they saw that they only needed an expression for x-2, so they stopped short and just replaced x-2 with 2xt/pi.

I do have to admit that what they did here is a little odd, not having technically solved for x, since they left both x and t in the expression. I imagine they did this because their main interest was to get rid of x-2, and they didn't mind leaving x there because they knew they could handle it. To actually solve for x, the normal thing to do would be to isolate the term with x, pi/x = pi/2 - t = pi/2 - 2t/2 = (pi - 2t)/2, and then take the reciprocal, x/pi = 2/(pi - 2t), then multiply by pi to get x = 2 pi/(pi - 2t). I imagine they saw that this would be unpleasant to work with (and maybe even did that at first and then changed their minds). It would have been nice if they explained their thinking in words, as I've mentioned. It would be more instructive that way.

I don't know what you mean by this. Nothing is cut, and there is no a/b.

What they did is to use the "cofunction identity", cos(pi/2 - u) = sin(u), to rewrite the expression using sin(t).

They also change a division by 2xt/pi into multiplication by pi/(2xt), and then split the limit of a product into a product of limits.

Thank you for answering.. I will try to learn/use the code after my exam, i wasn't thinking of doing so with this picture because well, my idea was to use colors to match the specific problems i had with the corresponding points in the picture. But i'll try to learn the code as soon as i can insha Allah.

As for the book not using words to explain, well.. they do but they're not in English, so i thought not to include them in the pics..

Thank you for your responses, i really appreciate them

 

[bbm25]

1) the limit has to be substituted with x. Not true. At the end you sub 2 for x
2) 0/0 cannot be an answer. Correct
3) (sin ax)/bx = a/b. NOT true at all. What is true is that the lim as x--> 0 of (sin ax)/bx = a/b

Thank you for clarifying!

 

[bbm25]

Isn't it clear that as x approaches 2, pi/2 - pi/x approaches pi/2 - pi/2, which is 0? That's the simplest kind of limit there is.

Actually i have to be honest.. It isn't clear.. I'm sorry.. But it isn't clear for me..

I also don't understand how they 'change the limit' from x approaching 0 to x approaching 2..

 

[bbm25]

As I said, they defined t as the difference between the argument of the cosine, pi/x, and the number it approaches, pi/2, as x approaches 0. This allows them to turn the limit into a limit as a variable (t) approaches 0.

I have to be honest.. I read this more than once and i still don't understand.. I'm sorry...

 

[Dr.Peterson]

Actually i have to be honest.. It isn't clear.. I'm sorry.. But it isn't clear for me.

But you yourself mentioned substitution. When a function is continuous at a point, you can find the limit by merely substituting x with its value. That's what I mean by the simplest kind of limit: just replace x with 2, and you get the limit.

I also don't understand how they 'change the limit' from x approaching 0 to x approaching 2..

The limit is never x approaching 2. It's t that approaches 0, just as we just showed.

I have to be honest.. I read this more than once and i still don't understand.. I'm sorry...

Then don't worry about it. As I've said repeatedly, they just told you to use that definition of t, so do it -- just make sure you follow what they did, not yet why. Once you follow the work, you will see why it was a useful choice. The fact that they gave the hint tells you that they didn't expect you to think of it yourself. It is not a typical problem, so this particular idea may never be needed again. But understanding their work may contribute to a gradual growth in your ability to get such ideas yourself.

But ultimately, they saw that the limit of sin(x)/x would be useful, so they did what they had to do to make it look like that.