limit / derivative question

[renegade05]

Just a quick couple questions:

\(\displaystyle \lim_{n \to \infty}(\frac{-2}{\pi})^n\)

I know this oscillates and approaches zero as n gets larger and larger. I forgot how to show this analytically though. Can someone please remind me. Is it If

\(\displaystyle \lim_{n \to \infty}(a)^n =0\) if \(\displaystyle a<0\) ?

Also, this is just out of curiousity, but: say \(\displaystyle y = (-2)^x\) what does \(\displaystyle \frac{dy}{dx} =?\)

Wolfram says : \(\displaystyle \frac{dy}{dx} =(-2)^x(ln(2)+i\pi)\)
Numberempire says: \(\displaystyle \frac{dy}{dx} =(-2)^xln(-2)\)

I am just wrapping up CALC II going into CALC III in sept. so it may be beyond my scope for now. I haven't learned about imaginary numbers yet or anything.

Intuitively, I would say \(\displaystyle \frac{dy}{dx} = DNE\) because the graph is discontinuous, but i would like to know for sure.
THANKS!

 

[tkhunny]

0 < 2/pi < 1 -- Why would intuition suggest that an oscillating series would fail to converge? Are you sure it's a < 0 and not |a| < 1?

 

[lookagain]

http://en.wikipedia.org/wiki/Limit_of_a_sequence


Scroll down to the 2nd item under "Examples"

 

[renegade05]

0 < 2/pi < 1 -- Why would intuition suggest that an oscillating series would fail to converge? Are you sure it's a < 0 and not |a| < 1?

You seem to have completely misread my post.

You are combing my two separate questions into one. I never said anything about intuition and convergence on the first one. And no I wasn't sure if it's a < 0 or |a| < 1 - hence me asking. I guess I should have made two separate posts for my two separate questions.

http://en.wikipedia.org/wiki/Limit_of_a_sequence


Scroll down to the 2nd item under "Examples"

I said I know the limit equals zero. I was asking for someone to explain how you would prove this analytically. I don't just like arbitrarily memorizing things without knowing the math behind it.

 

[lookagain]

\(\displaystyle \lim_{n \to \infty}(\frac{-2}{\pi})^n\)

I know this oscillates and approaches zero as n gets larger and larger.

I said I know the limit equals zero.

*No*, you did \(\displaystyle not\) state that. Look at the quote above.

Claiming that you "know this oscillates and approaches zero as n gets
larger and larger" is meaning that you "know the limit equals zero"
does not follow. It may indeed follow mathematically, but I could
not be expected for you to know it, as you typed different sentences
to describe the actions in the graph.

You don't get credit for it. A person who tells me what is in the first quote
box has not reasonably convinced me of what you claimed in the second
quote box. A person could think that it would not have a limit, given
that it oscillates.

If you had stated verbatim in the beginning of your post what you stated
in the second quote box (claiming you know the limit is zero), *then* you
would have been rightfully annoyed.

 

[renegade05]

\(\displaystyle \lim_{n \to \infty}(\frac{-2}{\pi})^n\)

I know this oscillates and approaches zero as n gets larger and larger.

I said I know the limit equals zero.

*No*, you did \(\displaystyle not\) state that. Look at the quote above.

Claiming that you "know this oscillates and approaches zero as n gets
larger and larger" is meaning that you "know the limit equals zero"
does not follow. It may indeed follow mathematically, but I could
not be expected for you to know it, as you typed different sentences
to describe the actions in the graph.

You don't get credit for it. A person who tells me what is in the first quote
box has not reasonably convinced me of what you claimed in the second
quote box. A person could think that it would not have a limit, given
that it oscillates.

If you had stated verbatim in the beginning of your post what you stated
in the second quote box (claiming you know the limit is zero), *then* you
would have been rightfully annoyed.

All my teachers use "approaches" and "equals" interchangeably when it comes to limits. Maybe one way is more correct than the other, but is it not very nit picky to call me out on that and not know what I was talking about? Maybe I'm wrong, but that's the way I was taught.

 

[tkhunny]

The expression or function approaches the limiting value.
The limit is (equals) the value.

It would be inappropriate to say the expression or function is (equals) the the limiting value.

 

[JeffM]

The history of calculus shows that it is very important to pick nits (which, if left unpicked, turn into lice).
(1) The function approaches a finite limit.
(2) If so, the limit equals some finite number.
But the function may or may not ever equal the limit.
Statements 1 and 2 express subtly different concepts, and it is sometimes important to keep them distinct. Being extra careful with the language that has been designed to express these concepts is a very useful habit although most of us are guilty of violating it at least occasionally.