Question

Help Please! Attachments to follow.

Answer 1

Here's what I got so far, but the answer is incorrect.

Answer 2

Hmm, looks like a cumbersome way to do it.
Do you know L'Hopital's rule?

Answer 3

I've heard of it...but i don't really understand it...

Answer 4

You differentiate the top and bottom, then reevaluate, it can get rid of indeterminate forms.

Answer 5

i'm not really sure how to do that...could you please show me?

Answer 6

Do you know what the derivative of cosine is?

Answer 7

yes. but i don't think i'm supposed to be using derivatives to find the limit...is there a way to just find the limit without using derivatives?

Answer 8

The hint given shows how to make a substitution to get it in terms of sine, and it looks like you did that correctly, but what is the value of a?

Answer 9

I was hoping you could tell me...

Answer 10

:-)
Let's look at the substitution made
cos(7x) = a*sin(x-3π/2)
Why do you suppose that works?

Answer 11

it could be derived from some trig identities that i'm unaware, possibly something like the double angle theorem?

Answer 12

*(oops, meant cos(7x) = a*sin(7(x-3π/2)) as the substitution)

Yes, it has to do that sine and cosine differ by 90º

Answer 13

So am i supposed to solve for a first?

Answer 14

Yes.

Pick some other value of x and compare cos(7x) with sin(7(x-3π/2)).

Answer 15

I set x = 1 and both cos(7x) and sin(7(x-3π/2)) came out the same...doesn't that imply that a = 1?

Answer 16

Are you sure they were exactly the same?

Answer 17

Oh! Whoops i must have typed it into my calculator wrong! They are opposites...so does that mean that a=-1?

Answer 18

Yes. :-)

Answer 19

So what's the limit?

Answer 20

The limit is -7!

Answer 21

You got it!

Answer 22

Thanks so much for your help! :D

Answer 23

Do some more research on L'Hopital's rule and try this problem using that method. I think you'll like it.

Answer 24

Is there any good resources that you'd recommend for it?

Answer 25

Wikipedia has a good article, but it might be a little too dense.
Try maybe Paul's Online Math Notes http://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx
It's the go-to place for most good notes and examples.

Answer 26

I'll show you how it works on the problem you had here:


\[\large \lim_{x \rightarrow 3π/2} \frac{cos(7x)}{x-3π/2} = \lim_{x \rightarrow 3π/2} \frac{\frac{d}{dx}cos(7x)}{\frac{d}{dx}(x-3π/2)}\]

\[\large \rightarrow \lim_{x \rightarrow 3π/2} \frac{-7sin(7x)}{1} = -7.\]

Answer 27

Better, eh?

Answer 28

Yeah much better! I will check out that site for sure! Thanks again! :D