I want to evaluate the following limit: \begin{eqnarray*} \lim_{t \rightarrow 0}M^{\prime\prime}_x(t) &=& \lim_{t \rightarrow 0} \frac{ 2t^{2} ( b^3e^{bt} - a^3e^{at} ) - 3t ( b^2e^{bt} - a^2e^{at} ) - ( be^{bt} - ae^{at} ) } {3(b-a)t^2} \\ \end{eqnarray*}
I plan to do so by using L'Hospital's rule. However, it appears to be a lot of work as I may have to apply the rule several times. I am thinking there should be a short cut. Any ideas?
Bob
Since the numerator is finite when $t=0$ and the denominator is zero, this expression diverges; no finite limit exists. Are you sure you have the right numerator?
At any rate, the shortcut is to expand the exponentials to about second order in $t$ and group terms with various powers of $t$. Which is how I saw that the expression will diverge: The constant and $t$ terms are non-zero.