Problem

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 \(\boxed{0}\).

From Solvely APP
Source: https://solvelyapp.com/problems/33238/

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