I have a question regarding an exponential function when time goes towards infinity. The equation I have is the following:
$R^3=\frac{1-b e^{\frac{(b-1)t}{a}}}{1-b}$ $\\ \\$ $equation 1$
where $e$ represents the exponential function. $a$, $b$ and $c$ are just constants.
Then the book tells me that when time goes to infinity, the above expression gives me: $R=(1-b)^{-\frac{1}{3}}$.
I find this very strange, since the numerator of equation 1 should not approach 1 when time goes to infinity. Does anyone understand why this is the case here?
The behavior of the given function as time 't' tends to infinity entirely depends upon the sign of the constant $\frac{(b-1)}{a}$. If $\frac{(b-1)}{a}$ is positive, then it is impossible to get a finite answer, as in your case, as and when time 't' tends to infinity.
However, if $\frac{(b-1)}{a}$ is somehow negative, then it is not very difficult to see how you would arrive at the result given in your textbook.
I, personally, believe it is an error of a minus sign which is rather common when not paying enough attention.