Finding Limit using Derivative

[Jules18]

Homework Statement

Express the limit as a derivative and evaluate.

lim_x-->pi/3 (Cos(theta) - 0.5)/(theta - pi/3)

The Attempt at a Solution

I derived the function using quotient rule and then didn't know what to do.

(theta - pi/3)(-Sin(theta)) - (Cos(theta) - 0.5)
_______________________________________

(theta - pi/3)²​

 

[Mark44]

Homework Statement

Express the limit as a derivative and evaluate.

lim_x-->pi/3 (Cos(theta) - 0.5)/(theta - pi/3)

The Attempt at a Solution

I derived the function using quotient rule and then didn't know what to do.

(theta - pi/3)(-Sin(theta)) - (Cos(theta) - 0.5)
_______________________________________

(theta - pi/3)²​

You're missing the point of this exercise. You are supposed to interpret this limit as the derivative of some function, not take the derivative of it.

Recall that the derivative can be defined as this limit:
$$\lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$$

Hint: cosine of what = .5?

 

[Jules18]

Okay, thanks

Does the wording of the question seem kind of weird to you? because I copied it word for word

 

[Mark44]

No, the wording seems fine to me.

BTW, you don't "derive" a function, you differentiate it. I know it seems weird that you don't "derive" something to get its derivative, but that's the way it is.

 

[Jules18]

I got -$$\sqrt{3}$$/2 as an answer, but I had to change the format of the derivative formula to this:

lim_theta-->pi/3 $$\stackrel{\underline{Cos(theta + h) - Cos(theta)}}{h}$$


But I don't know how to do it in the other format. Do you use conjugates, or what?

 

[Mark44]

Your answer is correct, so it seems you understand what the problem is asking for. Your revised limit needs some work, though, as it needs to be in terms of h going to 0, like so:
$$\lim_{h \rightarrow 0} \frac{cos(\pi /3 + h) - cos(\pi /3)}{h}$$
which equals -sin($\pi$ /3).

 

[n!kofeyn]

That's the correct answer, but I don't think that's what they're asking you to do, or at least how they want you to do the problem. Look at what Mark44 posted originally. That limit is equal to f'(a), which is f'(x) evaluated at a. Follow his hint to do it the original way. Either that or follow what he's posted above.

 

[Jules18]

yeah sry Mark that's what I meant.

Anyways, I'm good on that one, but I'm still a little fuzzy on how to find this limit:

lim_x-->(pi/3) $$\stackrel{\underline{(Cos(\theta) - 0.5)}}{(\theta - \pi/3)}$$


and I'm pretty sure that's what the question was asking.

 

[Mark44]

This is one version of the definition of d/d(theta)[cos(theta)], evaluated at theta = pi/3. Your limit should be as theta --> pi/3. No x.

 

[Jules18]

w/e about the x's and theta's that's just semantics.
I know that it's a version of d/d(theta). I want to know how to find the limit using that.

 

[n!kofeyn]

w/e about the x's and theta's that's just semantics.
I know that it's a version of d/d(theta). I want to know how to find the limit using that.

Well whether it's x or theta there changes the meaning of the problem completely. It's just a formula, but you have to sort of work backwards. An alternate definition of the derivative, which Mark44 already gave you, evaluated at the point a is
$$f'(a) = \lim_{\theta\to a} \frac{f(\theta)-f(a)}{\theta-a}$$
You should be able to evaluate your limit from here.

 

[Mark44]

w/e about the x's and theta's that's just semantics.

Doing well in mathematics and science and engineering is partly about paying attention to details ("semantics").

I know that it's a version of d/d(theta). I want to know how to find the limit using that.

Can you evaluate this limit?
$$\lim_{h \rightarrow 0} \frac{cos(\pi /3 + h) - cos(\pi /3)}{h}$$

You can convert the other limit to this form by letting h = $\theta - \pi /3$. Theta approaching pi/3 is equivalent to h approaching 0 in the limit above.

 

[Jules18]

Okay, thanks guys, sry if I sound frustrated

Except I had already made it to that step just by reading the question. It gave it to me in this format:

$$lim_{x->\pi /3} \stackrel{\underline{Cos(\theta ) - 0.5}}{\theta - \pi /3}$$

My problem isn't converting it to this, it's how to work with this formula to find the limit.
It might seem obvious to you guys, but I'm missing some kind of technique because everytime I end up with 0/0 and I confuse myself.

Help

 

[Mark44]

I think you are not realizing that cos(pi/3) = .5

 

[Jules18]

I am realizing that. That's why when I plug pi/3 into theta I get

(0.5 - 0.5)/(pi/3 - pi/3)

both are zero, so I get 0/0 and I freak out

 

[n!kofeyn]

$$\begin{align*}\lim_{\theta\to\pi/3} \frac{\cos\theta-0.5}{\theta-\pi/3} &= \lim_{\theta\to\pi/3} \frac{\cos\theta-\cos(\pi/3)}{\theta-\pi/3} = (\cos\theta)' \,\big|_{\theta=\pi/3} \\<br /> &= -\sin\theta \,\big|_{\theta=\pi/3} = -\sin(\pi/3) \\<br /> &= -\frac{\sqrt{3}}{2}\end{align*}$$

 

[Jules18]

Okay I get it.

Thanks so much guys, sry

 

[n!kofeyn]

Okay I get it.

Thanks so much guys, sry

No problem. What you were missing was using the definition of derivative that was posted to evaluate the limit. You kept trying to explicitly evaluate the limit instead of applying the definition in reverse.

 

[Mark44]

I am realizing that. That's why when I plug pi/3 into theta I get

(0.5 - 0.5)/(pi/3 - pi/3)

both are zero, so I get 0/0 and I freak out

This is why you need limits, because you can't directly evaluate the quotient.