http://rogercortesi.com/eqn/tempimagedir/eqn2028.png

is the answer really infinite? or is it 0?

and why?

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- Mar 20th 2007, 05:23 AM^_^Engineer_Adam^_^Limit Problem
http://rogercortesi.com/eqn/tempimagedir/eqn2028.png

is the answer really infinite? or is it 0?

and why? - Mar 20th 2007, 05:32 AMThePerfectHacker
Note,

(4-t^2)/(sqrt(4-t^2))=1/sqrt(4-t^2)

And when you approach t-->2 you have by substitution 1/0 which is an infinite limit. - Mar 20th 2007, 05:41 AM^_^Engineer_Adam^_^
ok .... but its indeterminate right?

why cant i use lhopitals? - Mar 20th 2007, 07:32 AMSoroban
Hello, ^_^Engineer_Adam^_^!

Quote:

Why cant I use L'Hopital?

. . . . . . . . . . . . . . (4 - x²)^½

We have: . f(x) .= .--------------

. . . . . . . . . . . . . . . 4 - x²

. . . . . . . . . . . . .½(4 - x²)^{-½}·(-2x)

Apply L'Hopital: . ------------------------- . = . ½(4 - x²)^{-½}

. . . . . . . . . . . . . . . . . .-2x

. . . . . . . . . . . . . . . . . . . 1

Now take the limit of: .----__-------__-

. . . . . . . . . . . . . . . . .2√4 - x²

- Mar 20th 2007, 09:14 AMJhevon
- Mar 20th 2007, 09:49 AMecMathGeek
Technically, 1/0 is not infinity, it's undefined. In other words, 1/0 has no value.

However, if we use limits, there are some situations where we might get division by zero when plugging a specific value into a function, but we get infinity when x approaches that value. In other words,

If f(x) = 1/Q(x), where Q(a)=0

Then f(a) = 1/0 = undifined (no solution)

However,

If lim(x->a+)f(x) = infinity

And lim(x->a-)f(x) = infinity

Then lim(x->a)f(x) = infinity

In other words, 1/0 never equals infinity, but the idea that 1/0 'is the same as' infinity comes from the fact that the limits of funtions when they approach some point, which are undefined at that point, approach infinity. - Mar 20th 2007, 10:01 AMJhevon
- Mar 20th 2007, 10:50 AMecMathGeek
I'm sorry if it sounds like I'm nit-picking; I'm not intending to. But the idea that "if a limit approaches 1/0 by definition it is an infinite limit" is not true. We are taught this in early calculus (and not really corrected in later calculus) because it's easy to remember, but as the value of the function approaches 1/0, the limit approaches infinity only under certain conditions: That the left and right hand limits of the function exist and are both equal to infinity. If any of these are not true, then 1/0 does not equal (approach) infinity.

for example:

lim(x->0) 1/x does not equal infinity because

lim(x->0+) 1/x = positive infinity

lim(x->0-) 1/x = negative infinity

There is no general limit of 1/x as x approaches 0. This is an example were as the function's value approaches 1/0, the limit does not exist (and for that reason is not equal infinity).

But I've gone WAY off topic, so just ignore everything I've said. ;) - Mar 20th 2007, 11:12 AMJhevon
- Mar 20th 2007, 11:50 AM^_^Engineer_Adam^_^
- Mar 20th 2007, 11:54 AMJhevon