I have two equations which represent the ratio of concentration to initial concentration in a chemical reactor, and I need to know what the limit of the left side would be, where when $t$ becomes large. The reaction under study is at steady state conditions. I know the right side is $1/(1+kT)$.
$$\lim_{t\to \infty}\left[\frac{2\big(a\ln{(t)}+b-c\ln{(t)}-d\big)\big(1-e^{-t/\tau}\big)}{a\ln{(t)+b}}\right]=\lim_{t\to \infty}\left[\frac{1}{1+k\tau}\left(1-e^{(-1/\tau+k)t}\right)\right]$$
You can re-write the expression as:
$$\lim_{t \rightarrow \infty }\frac{(2a - 2c) (1-e^{-t/{\tau}})\ln t}{a \ln t + b} + \lim_{t \rightarrow \infty } \frac{(2b-2d)(1-e^{-t/{\tau}})}{a \ln t + b}$$
First limit is $\frac{\infty}{\infty}$ form and could be solved using L'Hopital Rule and the second one goes to zero.