Find the limit of the sequence, using L'Hôpital's rule when appropriate. \[ \frac{n^{2}}{8^{n}} \]
Solution
Step 1 ：We are given the sequence \(\frac{n^{2}}{8^{n}}\) and asked to find its limit as n approaches infinity.
Step 2 ：The sequence is of the form \(\frac{P(n)}{Q(n)}\) where \(P(n)\) and \(Q(n)\) are functions of n. The limit of the sequence as n approaches infinity is the same as the limit of the ratio of the functions as n approaches infinity.
Step 3 ：Since the degree of the function in the denominator is greater than the degree of the function in the numerator, we can infer that the limit of the sequence as n approaches infinity is 0.
Step 4 ：To confirm this, we can use L'Hôpital's rule which states that the limit of a ratio of two functions as x approaches a certain value is equal to the limit of the ratio of their derivatives.
Step 5 ：We find the derivatives of \(n^{2}\) and \(8^{n}\) to be \(2n\) and \(8^{n}\log(8)\) respectively.
Step 6 ：We then find the limit of the ratio of these derivatives as n approaches infinity, which is 0.
Step 7 ：This confirms our initial thought that the limit of the sequence as n approaches infinity is 0.
Step 8 ：Final Answer: The limit of the sequence \(\frac{n^{2}}{8^{n}}\) as n approaches infinity is \(\boxed{0}\).
