13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{ⁿ√(e³ⁿ⁺⁴)}

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Identify the general term of the sequence: \(a_n = \sqrt[n]{e^{3n + 4}}\).

Rewrite the nth root expression using exponent properties: \(a_n = \left(e^{3n + 4}\right)^{\frac{1}{n}}\).

Simplify the exponent by distributing the \(\frac{1}{n}\): \(a_n = e^{\frac{3n + 4}{n}}\).

Separate the fraction in the exponent: \(a_n = e^{3 + \frac{4}{n}}\).

Analyze the limit as \(n \to \infty\): since \(\frac{4}{n} \to 0\), the limit of \(a_n\) is \(e^3\).

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주요 개념

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Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges. Understanding limits helps determine the long-term behavior of sequences.

Properties of Exponential and Root Functions

Exponential functions grow or decay at rates depending on their exponents, while root functions (like nth roots) can be rewritten using fractional exponents. Simplifying expressions involving roots and exponentials often involves rewriting nth roots as powers with exponent 1/n to analyze limits effectively.

Limit Laws and Simplification Techniques

Limit laws allow the separation and simplification of complex expressions into manageable parts. For sequences involving powers and roots, rewriting terms using exponent rules and applying limits to each component helps find the overall limit. Recognizing dominant terms is key to evaluating limits correctly.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{ⁿ√(e³ⁿ⁺⁴)}