7. Computing Limits

d. Limits at Infinity

2. Limit Laws

a. Examples

In general, the process of finding the limit of a function is to 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 \(\lim\limits_{x\to\infty} \dfrac{4+\dfrac{5}{x}-\dfrac{2}{x^2}}{3+\dfrac{4}{x}}\).

We use two column format to give the reasons:

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