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

{√(n² + 1) − n}

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Start by examining the sequence given: $a_n = \sqrt{n^2 + 1} - n$.

To find the limit as $n$ approaches infinity, consider rationalizing the expression by multiplying and dividing by the conjugate: $\sqrt{n^2 + 1} + n$.

Rewrite the sequence as $a_n = \frac{(\sqrt{n^2 + 1} - n)(\sqrt{n^2 + 1} + n)}{\sqrt{n^2 + 1} + n} = \frac{n^2 + 1 - n^2}{\sqrt{n^2 + 1} + n} = \frac{1}{\sqrt{n^2 + 1} + n}$.

Analyze the denominator $\sqrt{n^2 + 1} + n$ as $n$ becomes very large. Since $\sqrt{n^2 + 1}$ behaves like $n$ for large $n$, the denominator grows approximately like \$2n$.

Therefore, the sequence behaves like $\frac{1}{2n}$ for large $n$, and you can conclude the limit by considering the behavior of this simplified form as $n$ approaches infinity.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Algebraic Manipulation for Limits

When evaluating limits involving expressions like √(n² + 1) − n, algebraic techniques such as rationalizing the expression by multiplying numerator and denominator by the conjugate help simplify the expression and reveal the limit more clearly.

Behavior of Functions as n Approaches Infinity

Understanding how functions behave as n grows large, especially dominant terms like n² inside a square root, is crucial. Recognizing that √(n² + 1) behaves like n for large n helps in approximating and finding the limit.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{√(n² + 1) − n}