Use L’Hopital’s rule to find the limit of this sequence
(n^100)/(e^n)
...If you do L'Hop. Rule it would take forever, right? You would always get an (e^n) at the bottom and will have to use the L'Hop. rule 100 times to find the limit...100*n^99, 9900n^98, and etc.
Is there a shortcut to find the limit?
Or, am I doing something way wrong?
Yes, you are right. Directly applying L'Hopital's rule 100 times to find the limit of the sequence (n^100)/(e^n) can be a tedious and time-consuming process. However, there is a shortcut that can be used to simplify the calculation.
To find the limit of the sequence (n^100)/(e^n), we can rewrite it as:
(n/e)^n * n^100
Now, let's consider the limit of this expression as n approaches infinity.
The term (n/e)^n goes to infinity as n approaches infinity, due to the exponentially increasing value of n compared to e.
On the other hand, the term n^100 remains finite and does not depend on e.
Hence, the overall limit of the sequence (n^100)/(e^n) as n approaches infinity is infinity.
Therefore, there is no need to apply L'Hopital's rule multiple times in this case.