3. Determine whether the sequence $$ a_{n}=\frac{x+3}{\ln x} $$ converges or diverges. If it converges, find its limit.

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1. We interpret the sequence as $$ a_n=\frac{n+3}{\ln n} $$ where $$ n $$ is a positive integer.
2. As $$ n \to \infty $$, the numerator $$ n+3 $$ grows linearly, while the denominator $$ \ln n $$ grows much slower (logarithmically).
3. To analyze the limit, consider

$$
   \lim_{n \to \infty} \frac{n+3}{\ln n}.
   $$

For large $$ n $$, the behavior of $$ n+3 $$ is essentially the same as $$ n $$, so we examine

$$
   \lim_{n \to \infty} \frac{n}{\ln n}.
   $$

4. The numerator increases without bound, and since the logarithm increases very slowly, the ratio tends to infinity.
5. Thus, we conclude that

$$
   \lim_{n \to \infty} \frac{n+3}{\ln n} = \infty,
   $$

meaning that the sequence diverges (specifically, it diverges to infinity).
The sequence $$ a_n = \frac{n+3}{\ln n} $$ diverges to infinity as $$ n $$ approaches infinity.

Determine whether the sequence converges or diverges. If it converges, find its
limit.

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Determine whether the sequence converges or diverges. If it converges, find its limit.