18. Sequences

b4. Limit Laws

a. Examples

In general, the process of finding the limit of a sequence using the Limit Laws is essentially the same as finding the limit at infinity of a continuous function. Primarily, we repeatedly apply the Limit Laws until we get down to the Special Limits. If there is a condition to check, assume it holds and check it at the end.

Compute \(\lim\limits_{n\to\infty}\dfrac {4+\dfrac{5}{n}-\dfrac{2}{n^2}}{3+\dfrac{4}{n}}\).

We use two column format to give the reasons:

\(\lim\limits_{n\to\infty}\dfrac{4+\dfrac{5}{n} -\dfrac{2}{n^2}}{3+\dfrac{4}{n}} =\dfrac {\lim\limits_{n\to\infty} \left(4+\dfrac{5}{n}-\dfrac{2}{n^2}\right)} {\lim\limits_{n\to\infty} \left(3+\dfrac{4}{n}\right)}\)	The limit of a quotient is the quotient of the limits provided the limit of the denominator is non-zero.
\(=\dfrac {\lim\limits_{n\to\infty}4 +\lim\limits_{n\to\infty}\dfrac{5}{n} -\lim\limits_{n\to\infty}\dfrac{2}{n^2}} {\lim\limits_{n\to\infty}3 +\lim\limits_{n\to\infty}\dfrac{4}{n}}\)	The limit of a sum is the sum of the limits. The limit of a difference is the difference of the limits.
\(=\dfrac {4+5\lim\limits_{n\to\infty}\dfrac{1}{n} -2\lim\limits_{n\to\infty}\dfrac{1}{n^2}} {3+4\lim\limits_{n\to\infty}\dfrac{1}{n}}\)	The limit of a constant sequence is the constant. The limit of a constant times a sequence is the constant times the limit of the sequence.
\(=\dfrac{4+5\cdot0-2\cdot0}{3+4\cdot0}\)	Special Limit:   \(\lim\limits_{n\to\infty}\dfrac{1}{n^p}=0\,\) for \(p=1,2\)
\(=\dfrac{4}{3}\)	Simplify

Notice that the limit of the denominator was \(3\), thereby satisfying the requirement of the quotient rule.