Final answer:
The limit of the expression (1 + 2/n)^7n as n approaches infinity evaluates to e^14, making the correct answer c) (lim_n to [infinity] e^14).
Step-by-step explanation:
The student is asking to evaluate the limit of the expression (1 + \(\frac{2}{n}\))^7n as n approaches infinity. This is a form of the exponential growth limit that is foundational in calculus, often represented as:
\(\lim_{{n \to \infty}} \left(1 + \frac{k}{n}\right)^{n} = e^{k}\)
As n approaches infinity, the given expression closely resembles the definition of the natural exponential function e. Hence, we replace k with 2, and the exponent 7n can be split into 7 * n, mirroring the format necessary for applying the limit definition of the exponential function:
\(\lim_{{n \to \infty}} \left(1 + \frac{2}{n}\right)^{7n} = e^{7 * 2} = e^{14}\)
Therefore, the correct answer is c) \(\lim_{{n \to \infty}} e^{14}\).