**Given:**

The sequence is
an=4n1+9n
.

**Definition used:**

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

**Result used:**

The sequence
{rn}
is converges to zero when
−1<r<1
.

That is,
limn→∞rn=0 if −1<r<1
.

**Calculation:**

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

Compute
limn→∞4n1+9n
.

Divide by the highest denominator power.

limn→∞an=limn→∞4n9n1+9n9n=limn→∞(49)n19n+9n9n

Redefine the terms of the expressions as follows:

limn→∞an=limn→∞(49)n(19)n+1=limn→∞(49)nlimn→∞(19)n+limn→∞(1)

Since
49
and
19
are less than one and by using the result of converges sequence, both the terms
(49)n
and
(19)n
are converges to zero.

Therefore, the value of
limn→∞an
is,

limn→∞an=limn→∞4n1+9n=00+1=01=0

Since
limn→∞an
is exist and by using the definition of the sequence, it can be concluded that the sequence is converges to the limit 0.

Therefore, the sequence is converges to the limit 0.