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

$a_{n}=\frac{3 \sqrt{n}}{\sqrt{n}+2}$

To find: Whether the sequence converges or diverges and to obtain the limit if the sequence converges.

The sequence converges and its limit is 3.

Given:

The sequence is an=3nn+2.

Definition used:

If an is sequence and limn→∞an exists then the sequence an is convergent otherwise it is divergent.

Calculation:

Obtain the limit of the sequence to investigate whether the sequence converges or diverges.

Compute limn→∞an=limn→∞3nn+2.

Divide the numerator and the denominator by n.

limn→∞an=limn→∞3nnn+2n=limn→∞3nn+2n

Redefine the terms of the numerator and the denominator as follows:

limn→∞an=limn→∞31+2n=limn→∞(3)limn→∞(1)+limn→∞2n

Apply the property limn→∞(kna)=0 and simplify the numerator and the denominator.

limn→∞an=31+2∞=31+0=31=3

Since limn→∞an exist and by using the definition of the sequence, it can be concluded that the sequence converges to the limit 3.

Therefore, the sequence converges to the limit 3.