How To Find The Limit Of A Polynomial Function As X Approaches Infinity: A Simplified Guide

lim x->∞ pow top > pow bottom

∞, -∞

In order to determine the limit of the function as x approaches infinity where the numerator is a higher degree polynomial than the denominator, we can use the rules of polynomial limits. Specifically, we can divide through by the highest degree term in the denominator and numerator to simplify the expression.

Let’s consider a general form of the given limit:

lim x -> ∞ (ax^n + bx^(n-1) + … + c) / (dx^m + ex^(m-1) + … + f)

Dividing both numerator and denominator by the highest degree term in the denominator (dx^m), we get:

lim x -> ∞ (ax^n / dx^m + bx^(n-1) / dx^m + … + c / dx^m) / (1 + ex^(-1) dx^(-m+1) + … + f / dx^m)

Now, we can take the limit as x approaches infinity, and the terms with powers of x in the denominator all approach zero, leaving us with:

lim x -> ∞ (ax^n / dx^m) / 1

as the limit.

Note that if the degree of the numerator is greater than the degree of the denominator, then the limit can be simplified to either infinity or negative infinity, depending on the sign of the leading coefficient of the numerator. Specifically, if the leading coefficient of the numerator is positive, then the limit approaches infinity, and if it is negative, then the limit approaches negative infinity.

Therefore, in the given limit where the numerator is a higher degree polynomial than the denominator, the limit will approach either positive or negative infinity depending on the sign of the leading coefficient of the numerator.

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