Use algebra to simplify the expression before evaluating the limit. In particular, factor the highest power of n from the numerator and denominator, then cancel as many factors of n as possible. If the sequence does not converge, enter diverges in the final answer box. limn→[infinity]2-7n+9n28n2+2n-6

a) {2/n²-7/n+9}/{8+2/n-6/n²}

b) 9/8

c) The sequence converges

Step-by-step explanation:

Given the limit of the function

limn→[infinity]2-7n+9n²/8n²+2n-6

To simplify the function given, we will have to factor out the highest power of n which is n² from the numerator and the denominator. The function will then become;

2-7n+9n²/8n²+2n-6

= n²{2/n²-7/n+9}/n²{8+2/n-6/n²}

The n² at the numerator will then cancel out the n² at the denominator to have resulting simplified equation as;

{2/n²-7/n+9}/{8+2/n-6/n²}

Evaluating the limit of the resulting equation will give;

limn→[infinity] {2/n²-7/n+9}/{8+2/n-6/n²}

Note that limn→[infinity] a/n = 0 where a is any constant.

Therefore;

limn→[infinity] {2/n²-7/n+9}/{8+2/n-6/n²}

= (0-0+9) / (8+0-0)

= 9/8

Since the limit of the sequence gives a finite value which is 9/8, thus the sequence in question is a convergent sequence.

The limit of a sequence only diverges if the limit of such sequence is an infinite value.