Let P and Q be polynomials. Find \lim \limits_x \to \infty} \frac {P(x)}{Q(x)} if the degree of P...

Question:

Let {eq}P {/eq} and {eq}Q {/eq} be polynomials. Find {eq}\lim \limits_{x \to \infty} \frac {P(x)}{Q(x)} {/eq} if the degree of {eq}P {/eq} is:

a. less than the degree of {eq}Q, {/eq}

b. greater than the degree of {eq}Q. {/eq}

Limit of a rational function:

The value of the limit of a rational function, depends on the value of the degree of the polynomials of the numerator and the denominator, there are three cases, we address the two cases we need in the exercise:

-If the degree of the numerator is greater than the degree of the denominator, the limit may be infinite or less infinite.

-If the degree of the numerator is less than the degree of the denominator, the limit is zero.

Answer and Explanation:

Part a.

If the denominator has a greater degree than the numerator, the numerator will cancel out completely leaving only a constant, but a remaining variable in the denominator that could not be canceled.

When taking the limit of this variable to infinity, this results in a constant divided by infinity, which is zero.

Therefore we have:

{eq}\displaystyle L = \lim \limits_{x \to \infty} \frac {P(x)}{Q(x)}\\ \textrm { The degree of P is less than the degree of Q,}\\ \displaystyle L =0 {/eq}

Part b.

If the numerator has a greater degree than the denominator, the denominator will cancel out in its entirety leaving only a constant, but a remaining variable in the numerator that could not be canceled.

When taking the limit of this variable to infinity, the function will also approach infinity.

Therefore we have,

{eq}\displaystyle L = \lim \limits_{x \to \infty} \frac {P(x)}{Q(x)}\\ \textrm { The degree of P is greater than the degree of Q,}\\ \displaystyle L =\infty {/eq}

This last case may be the limit also less infinite, when the coefficient that accompanies the variable with the highest degree of P, is negative.

There is another case, the degree of the two polynomials is the same, in this case the limit is the quotient of the coefficients of the variable of greater degree.


Learn more about this topic:

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Rational Function: Definition, Equation & Examples

from GMAT Prep: Help and Review

Chapter 10 / Lesson 11
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