Question

A_n=5 n²+14 n/3 n⁴-5 n²-22, b_n=5/3 n²

Calculate the limit. (Give an exact answer. Use symbolic notation and fractions where needed.

Answer 1

The limit of the sequence a_n as n approaches infinity is 5/3, and the limit of the sequence b_n as n approaches infinity is 0.

Step-by-step explanation:

The question is asking for the limit of the sequence a_n as n approaches infinity. Given a_n = (5n² + 14n) / (3n´ - 5n² - 22) and b_n = 5/(3n²), let's find the limit of each sequence separately.

For sequence a_n, as n approaches infinity, the highest power of n dominates both the numerator and denominator. Thus, we divide both by n´, the highest power in the denominator, to find the leading coefficient, which is 5/3 in this case. So, the limit of a_n is 5/3.

For sequence b_n, as n approaches infinity, the term in the denominator grows much larger than the constant in the numerator, which leads to the limit being 0.