I just finished a question on indeterminate powers and I'm fed up with how slow and inefficient my method of evaluating limits at discontinuities and infinities are.
The question was $\lim_\limits{x\rightarrow\infty}\left(1+\dfrac{a}{x}\right)^{x}$. After using a few limit laws, I was able to get to the correct answer. Which is:
$\lim_\limits{x\rightarrow\infty}\left(1+\dfrac{a}{x}\right)^{x}=e^a$
I won't show all of my work here, but, I was able to get the expression into the form,
$\lim_\limits{x\rightarrow\infty}[f(x)]^{g(x)}= e^{\lim_\limits{x\rightarrow \infty}\frac{\ln(1+\frac{a}{x})}{x^{-1}}}$
I then used L’Hôpital's rule to get rid of that nasty indeterminate exponent and was able to get the exponent to look like this:
$\lim_\limits{x\rightarrow\infty}\frac{ax}{(x+a)}$ $\leftarrow$That was easy :)
My problem is, I don't know an efficient way of evaluating the limit as it approaches an infinity (or some discontinuity). So evaluating the limit of this exponent was a real pain for me. For about 4 years, to solve limits I've made a mental table of values and subbed in a few $x$-values towards pos/neg infinity, keeping track of the results of the outputs, then guesstimating which $y$-value they approach. If I'm lucky, it's a nice number... but usually its an irrational number or a crazy fraction:(
If anyone knows a more efficient way of evaluating limits, or even, if you evaluate limits a different way; I would love to know. Thanks a million!