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System Message

[00:00:00] *** Please note: All sessions are recorded for quality control. ***

Saori (Customer)

[00:00:00] lim(n->infinity)
cos(n*pi)/n

[00:00:05] hello

Richard H (Tutor)

[00:00:08] Hello, and welcome to Tutor.com

Saori (Customer)

[00:00:22] The question is asking to
evaluate the given limit

Richard H (Tutor)

[00:00:34] Ok, I see. Its like this, right?

Saori (Customer)

[00:00:50] yes

Richard H (Tutor)

[00:01:10] Ok, just making sure. Are there any methods we're supposed to use or any methods that you're going over in class at the moment

Saori (Customer)

[00:01:42] not specifically

[00:02:06] I learned to divide the
denominator by the highest
degree variable

[00:02:13] so for example, divide n to
everything

[00:02:20] but I dont think this works on
this problem

Richard H (Tutor)

[00:02:41] Indeed, it won't unfortunately. Give me a minute to finish checking it over, then I'll get right back to you

[00:04:15] Ok, I think I got it. What do you think on this problem?

Saori (Customer)

[00:04:44] well, I dont know how to work
with cosines in limits

Richard H (Tutor)

[00:05:19] I see what you mean, but don't let it intimidate you. sines and cosines have a few properties that are very ... nice.

[00:05:56] When I see this problem, another function comes to mind [the connection will be seen in a moment]. That is 1/n. What is the limit as n-->infinity of (1/n) ?

Saori (Customer)

[00:06:10] 0

Richard H (Tutor)

[00:06:14] Indeed!

[00:06:28] Now, we can use that to help us in this problem. How? Well:

[00:06:55] we have cos(n*pi) there, but cosine has a certain range of values it can take, regardless of the angle inside. What is the range of cosine?

Saori (Customer)

[00:07:03] -1 to 1

Richard H (Tutor)

[00:07:08] Correct!

[00:07:20] We can use that to our advantage. Like this:

[00:08:27] Since the limit of the functions -1/n and +1/n are each zero, by the Squeeze Theorem:

Saori (Customer)

[00:08:56] So cos and sin does not have
definite limit?

Richard H (Tutor)

[00:09:17] Well, no, because they oscillate between -1 and +1

Saori (Customer)

[00:09:21] ok

Richard H (Tutor)

[00:09:35] What does this mean for our given limit?

Saori (Customer)

[00:09:47] 0?

Richard H (Tutor)

[00:10:04] Indeed!

Saori (Customer)

[00:10:11] oh I see

[00:10:25] what would the limit be for a
tangent?

Richard H (Tutor)

[00:11:04] Well, tangent repeats itself but its range is all real numbers. Since it repeats itself, however, it has no specific limit

Saori (Customer)

[00:11:12] ok

Richard H (Tutor)

[00:11:30] However, its range is not -1 to +1, its all real numbers, so the same trick wouldn't work for tan(n*pi) on top

Saori (Customer)

[00:11:42] ok I see

[00:11:49] I got it! Thanks!

Richard H (Tutor)

[00:11:59]
Do you have any other questions?

Saori (Customer)

[00:12:05] no that's all

[00:12:13] thank you!

Richard H (Tutor)

[00:12:18] Cool! If you have any other questions, anytime, log in again!

[00:12:21] Glad to have been of help

[00:12:24] Have a great day!

Saori (Customer)

[00:12:27] you too!

Richard H (Tutor)

[00:12:32] Thanks. Bye