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Limits Involving Polynomials at Infinity

April 8th, 2009 |

Author: Merrin

Rational functions are continuous in their domain. Possible tricky limits of rational functions are the behaviors at positive and negative infinity or when the denominator equals zero. Calculating the limit of a rational function as x approaches infinity can always be done by the same method.

For a rational function, P(x)/Q(x), the highest degree terms of P(x) and Q(x) determine

$\mathop{\lim }\limits_{x\to \infty } R(x)=\mathop{\lim }\limits_{x\to \infty } \cfrac{P(x)}{Q(x)}$

There are three possibilities. The degree of P(x) is less than, equal to, or greater than the degree of Q(x). Here we show what to expect for all three examples. Also we use the fact that

$\mathop{\lim }\limits_{x\to \infty } \cfrac{1}{x^{a} } =0, a>0$

For a general rational function, we can take the limit as x approaches infinity.

${R(x)}={\cfrac{a_{n} x^{n} +a_{n-1} x^{n-1} +...+a_{0} }{b_{m} x^{m} +b_{m-1} x^{m-1} +...+b_{0} } }$

If n > m then

$\mathop{\lim }\limits_{x\to \infty } x^{n-m} =\infty$

if n < m then

$\mathop{\lim }\limits_{x\to \infty } x^{n-m} =0$

and if n=m then

$\cfrac{a_{n} }{b_{n} } \mathop{\lim }\limits_{x\to \infty } x^{n-m} =\, \cfrac{a_{n} }{b_{n} }$

Example 1. For the following rational function find the limit at infinity.

$\cfrac{P(x)}{Q(x)} =\cfrac{x^{2} +5}{x^{3} +x^{2} +9}$

Solution 1. Factor the highest degree terms and then take the limit as x approaches infinity.

${\mathop{\lim }\limits_{x\to \infty } \cfrac{P(x)}{Q(x)} } = {\mathop{\lim }\limits_{x\to \infty } \cfrac{x^{2} (1+5x^{-2} )}{x^{3} (1+x^{-1} +9x^{-3} )} }$
$= {\mathop{\lim }\limits_{x\to \infty } \cfrac{1}{x} \mathop{\lim }\limits_{x\to \infty } \cfrac{(1+5x^{-2} )}{(1+x^{-1} +9x^{-3} )} }$
$= {0\cdot \cfrac{1+0}{1+0+0} =0}$

This function has a horizontal asymptote at y=0.

Example 2. For the following rational function find the limit at infinity.

$\cfrac{P(x)}{Q(x)} =\cfrac{3x^{4} +2x^{2} }{x^{4} +2x^{2} +5}$

Solution 2. Factor the numerator and denominator then take the limit at infinity.

${\mathop{\lim }\limits_{x\to \infty } \cfrac{P(x)}{Q(x)} } = {\mathop{\lim }\limits_{x\to \infty } \cfrac{3x^{4} (1+2x^{-2} )}{x^{4} (1+2x^{-2} +5x^{-4} )} }$
$= {\mathop{\lim }\limits_{x\to \infty } 3\cfrac{x^{4} }{x^{4} } \mathop{\lim }\limits_{x\to \infty } \cfrac{(1+2x^{-2} )}{(1+2x^{-2} +5x^{-4} )} =3\cdot \cfrac{1+0}{1+0+0} =3}$

This function has a horizontal asymptote at y=3.

Example 3. For the following rational function find the limit at infinity.

$\cfrac{P(x)}{Q(x)} =\cfrac{x^{6} +5x^{4} }{x^{3} +3x^{2} +1}$

Solution 3. Factor the numerator and denominator then take the limit. The degree of P(x) is larger than Q(x).

${\mathop{\lim }\limits_{x\to \infty } \cfrac{P(x)}{Q(x)} } = {\mathop{\lim }\limits_{x\to \infty } \cfrac{x^{6} +5x^{4} }{x^{3} +3x^{2} +1} }$
$= {\mathop{\lim }\limits_{x\to \infty } \cfrac{x^{6} }{x^{3} } \mathop{\lim }\limits_{x\to \infty } \cfrac{(1+5x^{-2} )}{(1+3x^{-1} +x^{-3} )} }$
$= {\infty \cdot \cfrac{1+0}{1+0+0} =\infty }$