Part 2/6:
\lim_{n \to \infty} \frac{n}{n + 1} = \lim_{n \to \infty} \frac{1}{1 + \frac{1}{n}}
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As ( n ) approaches infinity, ( \frac{1}{n} ) approaches zero. Thus, this limit simplifies to:
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\frac{1}{1 + 0} = 1
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This fundamental result shows that as ( n ) becomes exceedingly large, the ratio ( \frac{n}{n + 1} ) approaches 1.
Limit Involving Logarithms
Next, we consider the limit of the sequence defined by ( \frac{\log n}{n} ) as ( n \to \infty ). This too produces an indeterminate form ( \frac{\infty}{\infty} ). To solve this, we can employ L'Hôpital's Rule, which allows us to differentiate the numerator and denominator:
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