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Re: A limit at a point means a function is continuous at that point,
Posted:
Nov 10, 2017 10:02 AM


L'Hopital doesn't fail for:
f(b)  f(a)  = ? for lim b>a b  a
Take a as a constant, then f(a) is also a constant. The derivatiev of a constant is 0, so we have:
d a/db = 0 d f(a)/db = 0
Now take the derivative of the numerator and denumerator, you get the following:
d(f(b)f(a))/db f'(b)  0  =  = f'(b) for lim b>a d(ba)/db 1  0
So we can find the limit of this zero by zero division, using L'Hopital rule, and we get:
f(b)  f(a)  = f'(b) for lim b>a b  a
Exercise: Use chain rule and other movements of a,b, what will be the result of limit?
Am Freitag, 10. November 2017 13:36:24 UTC+1 schrieb John Gabriel: > On Friday, 10 November 2017 00:15:46 UTC5, George Cornelius wrote: > > > > Somewhere else I wrote 2*e^(1/x^2)/x^3 = 1. Sorry for > > > that error, but in fact it is also: > > > > > > lim x>0 2*e^(1/x^2)/x^3 = 0 > > > > > Which is a nice little limit exercise. > > > > The exponential function has an "essential singularity" at infinity. > > There is nothing "essential" in mathematics and "at infinity" is an impossible scenario. We simply say the function is undefined at x=0 and avoid use of words such as singularity which morons like Stephen Hawking have overused. > > > > > I have long forgotten my complex analysis but was going to guess that > > that implied that it grows near infinity faster than any power > > of x, and thus x^(3)/e^(1/x^2) goes to zero as x goes to zero. > > There is no limit as I have proved. You cannot substitute a nonnumber that is, oo into e^x. e^(oo) is meaningless nonsense. oo is not a limit. > > > > > But what do I know? L'Hopital is a bit tricky, but using > > You can't use L'Hopital's property unless it has the form oo/oo or 0/0 which is not case here. Also L'Hopital fails for several functions which do have those forms. > > > the limit of the log of the function gives you a straightforward > > solution for x>0+ . For the other side of the limit you want > > (x)^3/e^(1/x^2) as x>0+, which will be the negative of the > > other limit and therefore zero as well. > > > > > Am Donnerstag, 9. November 2017 14:42:37 UTC+1 schrieb Zelos Malum: > > >> >The idiotic mainstream tendency is to transfer the limit to the exponent, that is, > > >> > > >> They do not > > >> > > >> >Then what do orangutans do? They say: > > >> > > > >> >0=e^(oo) > > >> > > >> Nope, for real numbers that is not defined or meaningful so they don't do that. Iti s your idiocy not theirs.



