murshid

Thanks, dave!

"What the Tortoise Said to Achilles" by Lewis Carroll: http://www.ditext.com/carroll/tortoise.html What exactly is wrong with the logic here? Is anything wrong at all?

Can anyone suggest a good book on Complex Analysis? I need a book that would be good for self studying.

. I have a question about the original problem: [math]\lim_{t\rightarrow 0^{+}} \frac{1-x^t}{t} = -\ln x[/math] for [math]x > 0[/math] Why is it [math]t \rightarrow 0^{+}[/math] instead of [math]t \rightarrow 0^{-}[/math] or [math]t \rightarrow 0[/math]? I don't see why it shouldn't work for either [math]t \rightarrow 0^{-}[/math] or [math]t \rightarrow 0[/math].

I have asked a question about the problem with the factorial sign here: http://www.scienceforums.net/topic/3751-quick-latex-tutorial/page__view__findpost__p__572135 .

Why doesn't the factorial sign work in LaTex here? for example, [math]\frac{1}{2}[/math] works, but when I use factorial sign after the '2' in the denominator, I get an error message: [math]\frac{1}{2!}[/math] .

I tried that. It doesn't work. For example, [math]\frac{\ln x}{1} [/math] works; but when I put a factorial sign after the '1' in the denominator, I get an error message: [math]\frac{\ln x}{1 !} [/math].

Can you tell me why the factorial sign isn't working in LaTex? .

I think I've got it now. The expansion of [math]x^t[/math] near [math]t = 0[/math] is: [math]1 + \frac{\ln x}{1}t + \frac{(\ln x)^2}{2}t^2 + \frac{(\ln x)^3}{6}t^3 + \cdots[/math] Therefore, [math] \lim_{t\rightarrow 0^{+}} \frac{1-x^t}{t} [/math] [math]= \lim_{t\rightarrow 0^{+}} \frac{- \frac{\ln x}{1}t - \frac{\left (\ln x\right )^2}{2}t^2 - \frac{\left (\ln x\right )^3}{6}t^3 - \cdots}{t} [/math] [math]= \lim_{t\rightarrow 0^{+}} - \frac{\ln x}{1} - \frac{\left (\ln x\right )^2}{2}t - \frac{\left (\ln x\right )^3}{6}t^2 - \cdots [/math] [math]= - \ln x [/math]

I managed to solve it using L'Hopital's rule. [math]\frac{d}{dt} x^{t} = x^t \ln x[/math] (I got it by letting [math]y = x^t[/math], which is equivalent to [math]\ln y = t \ln x[/math], and then differentiating both sides with respect to t). But what did you mean by Series expand [math]x^{t}[/math]? Did you mean the Taylor/Maclaurin Series expansion? I have been out of touch with calculus for the last few years. So it would really help if you could give me the series expansion of [math]x^{t}[/math] about [math]t = 0[/math].

In the preface to William Dunham's book "Euler: The Master of Us All", he wrote, [math]\lim_{t\rightarrow 0^{+}} \frac{1-x^t}{t} = -\ln x[/math] for [math]x > 0[/math] Can anyone tell me how he got that result?

Thanks for the replies!

Are the ideas that "Higgs boson exists" and "supersymmetry is a symmetry of nature" falsifiable even in principle? If yes, then how?