Here is a useful equation:
.
For your problem, instead of evaluating the limit of the ratio, evaluate the ratio of the limits:
The limit in the denominator is 1. Division by 1 won't change anything, so the answer to your original problem is just the limit in the numerator.
If we let r = 2 and x = n + 1, then the limit in your problem becomes . Remember the useful equation at the top?
The limit is just e^r. Substituting r = 2 gives: e^2.
Thank you so much , it makes sense now.
My mistake was not evaluating the numerator correctly. I found the limit of the denominator correctly, but for the numerator I said that the limit when n approaches infinity was = (1)^ infinity which is the same as 1. What was wrong with this reasoning?
is what is called an indeterminate form. It's value cannot be determined. In this problem, it turns out it is "equal" to e^2. In another problem, it might turn out to be "equal" to sin(1). Some other common indeterminate forms are and . Here is a link to a Wolfram article on indeterminate forms: Indeterminate -- from Wolfram MathWorld