Identities like these...
... are not consequences of some particular theorem but only a question of elementary 'good sense'...
Kind regards
1. lim as x goes to 2...(x^2 - 4)/(x-2)
2. lim as x goes to 0...sin 2x + x^2 cos 5x
3. lim as x goes to 0 from the negative side... [(square root)x^2]/x
as ive said, ive gotten all the limits, but I need to tell which theorem i used.... we can only use like the 5 most basic ones, and i cant see how any of them apply to any of these
thats exactly what i thought, but they explicitly say to note what limit theorems you are using
the ones we got are:
basically the 4 operations using limits f and g...eg lim f(x) + g(x)= lim f(x) + lim g(X)
squeeze principal
continuity of indefinite inegrals
(x^2 - 4)/(x-2) = (x-2)(x+2)/(x-2) . Now, from the very definition of limit, when x --> 2 we explicitly rule out the value x = 2, so that above we can divide by x-2 and get very simply lim (x + 2) = 4 when x --> 2
For the enxt one you use arithmetic of limits: both sin(2x) and x^2*cos 5x have limit zero when x --> 0: the first one directly by continuity of sin x. and the second one because x^2 --> 0 clearly and cos 5x is bounded.
The last one is nice: since we're working with negative x's, Sqrt(x^2) = |x| ==> the limit wanted is lim |x|/x = lim (-1) = -1 when x --> 0-
Tonio
Theorem: if f(x)= g(x) for all x except x= a, then .
Theorem: if f is continuous at x= a, then2. lim as x goes to 0...sin 2x + x^2 cos 5x
Well, more the definition of "continuous" than a theorem. And sine and cosine are continuous for all x.
which is -x for x negative. That is, which is -1 as long as x is not 0. Now use the theorem I cited for (1).3. lim as x goes to 0 from the negative side... [(square root)x^2]/x
as ive said, ive gotten all the limits, but I need to tell which theorem i used.... we can only use like the 5 most basic ones, and i cant see how any of them apply to any of these
The post by HoI (which you quoted) is not an opinion, it is mathematical fact. Details like this are important. It is never a waste of time to note and correct errors of understanding. These sorts of errors are extremely common - it is not a waste of time to make sure that the OP has the correct mathematical understanding. Far from being tedious, the discussion in this thread is extremely valuable.
Edit: Thread closed. The discussion was getting off-point. As far as I can see, the questions have been satisfactorily answered. twostep08, if you require more help please pm me.