Find the limit if it exists.
[tex]\lim _{n \rightarrow \infty}\left(\frac{1}{n}+1\right)^{3 n}[/tex]
A. [tex]$3 e$[/tex]
B. [tex]$\frac{1}{e^3}$[/tex]
C. [tex]$e^3$[/tex]
D. [tex]$\frac{1}{3 e}$[/tex]
Answer (2)
Recognize the limit as a variation of the standard limit lim n → ∞ ( 1 + n 1 ) n = e .
Rewrite the given expression as lim n → ∞ (( 1 + n 1 ) n ) 3 .
Apply the limit to obtain ( lim n → ∞ ( 1 + n 1 ) n ) 3 = e 3 .
The final answer is e 3 .
Explanation
Problem Analysis We are asked to find the limit of the expression lim n → ∞ ( n 1 + 1 ) 3 n . This is a limit problem that can be solved using the knowledge of standard limits.
Using Known Limits We know that lim n → ∞ ( 1 + n 1 ) n = e . We can rewrite the given expression to match this form.
Rewriting the Expression We can rewrite the expression as follows: n → ∞ lim ( n 1 + 1 ) 3 n = n → ∞ lim ( 1 + n 1 ) 3 n = n → ∞ lim ( ( 1 + n 1 ) n ) 3
Applying the Limit Now, we can apply the limit: n → ∞ lim ( ( 1 + n 1 ) n ) 3 = ( n → ∞ lim ( 1 + n 1 ) n ) 3 = e 3
Final Answer Therefore, the limit of the given expression as n approaches infinity is e 3 .
Examples
In population modeling, the expression ( 1 + n 1 ) kn can represent the growth of a population over time, where n is the number of time intervals, and k is a growth factor. As the number of intervals becomes very large, the limit of this expression gives the continuous growth rate, which is e k . This is useful in understanding how populations or investments grow continuously over time.
The limit lim n → ∞ ( n 1 + 1 ) 3 n equals e 3 . Therefore, the correct answer is option C: e 3 .
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