What is the parametrization of the graph of ln(x)?

[lovexmango]

Let k(x) be the curvature of y=ln(x) at x. Find the limit as x approaches to the positive infinity of k(x). At what point does the curve have maximum curvature?

You're supposed to parametrize the graph of ln(x), which I found to be x(t)=(t,ln(t)). And you're not allowed to use the formula with the second derivative, only k(t)=magnitude T'(t)/ magnitude v'(t).
I have problem simplifying the formula for T'(t) and k(t).

 

[HallsofIvy]

If x= (t, ln(t)) then v= (1, 1/t) so |v| is \sqrt{1+ 1/t^2} and T= (\frac{1}{\sqrt{1+ 1/t^2}}, \frac{1}{t^2\sqrt{1+ 1/t^2}}).

To simplify t^2\sqrt{1+ 1/t^2}, take one of the "t^2" inside the square root: t\sqrt{t^2+ 1}. You can simplify the first component by multiplying both numerator and denominator by t: \sqrt{1}{\sqrt{1+ 1/t^2}}= \frac{t}{t\sqrt{1+ 1/t^2}}= \frac{t}{\sqrt{t^2+ 1}}. So T= (\frac{t}{\sqrt{t^2+ 1}}, \frac{1}{t\sqrt{t^2+ 1}})