# What is the arclength of f(t) = (tlnt,t/lnt) on t in [1,e]?

Notice that ${\lim}_{t \rightarrow {1}^{+}} \frac{t}{\ln} t = \infty$, since $\ln \left(1\right) = 0$. This means that the arc length is infinite, as the function is asymptotic at $t = 1$.
We can also note that if we attempt to apply the arc length formula $L = {\int}_{a}^{b} \sqrt{{\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}$, the integral diverges.
Attached is a plot of the parametric graph for $t \in \left(1 , e\right]$: