Answer:
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.
