Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.
A) $\frac { 1 } { 6 }$
B) $- 5$
C) $- \frac { 5 } { 6 }$
D) Diverges

Question

Quizplus · Accepted Answer

To determine if the sequence converges, we can take the limit as $n$ approaches infinity. 
$$\lim_{n 	o \infty} a_n = \lim_{n 	o \infty}\frac{n^2-5}{6n^2+1}$$
Divide both the numerator and denominator by $n^2$:
$$\lim_{n 	o \infty}\frac{1-\frac{5}{n^2}}{6+\frac{1}{n^2}}$$
As $n$ approaches infinity, the terms $\frac{5}{n^2}$ and $\frac{1}{n^2}$ approach zero. Therefore, we have:
$$\lim_{n 	o \infty}\frac{1-0}{6+0} = \frac{1}{6}$$
Thus, the sequence converges to $\frac{1}{6}$. The correct answer is $\boxed{	extbf{(A)}}$.

Determine Whether the Sequence Defined By an=n2−56n2+1a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }an​=6n2+1n2−5​

Determine Whether the Sequence Defined By $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$