Question

Prove that if \(p(x)=a_{n} x^{n}+\cdots+a_{1} x+a_{0}\) and \(q(x)=b_{m} x^{m}+\cdots+b_{1} x+b_{0}\left(a_{n} \neq 0, b_{m} \neq 0\right),\) then \(\lim _{x \rightarrow \infty} \frac{p(x)}{q(x)}=\left\\{\begin{array}{ll}0, & nm\end{array}\right.\)

Short answer

The task provides a fundamental aspect of limit of a ratio of 2 polynomials when \(x\) approaches infinity. The exact behavior of such limit depends on the relative degrees of these polynomials: If the denominator has a higher degree, the limit is 0; If they have the same degree, the limit becomes the division of the leading coefficients; if the numerator has a higher degree, the limit goes to \(+-\infty\) depending to their leading coefficients.

Step-by-step solution

Case 1: Degree of \(p(x)\) is less than degree of \(q(x)\)

If \(n<m\), then the degree of \(q(x)\) is higher than that of \(p(x)\). As \(x\) approaches infinity, \(q(x)\) will thus increase faster than \(p(x)\). Therefore, \(\lim _{x \rightarrow \infty} \frac{p(x)}{q(x)} =0\). The ratio of a smaller degree polynomial to a larger degree polynomial tends to 0 as \(x\) approaches infinity.

Case 2: Degree of \(p(x)\) equals degree of \(q(x)\)

If \(n=m\), then the degree of \(p(x)\) equals the degree of \(q(x)\). As \(x\) approaches infinity, the leading terms will dominate since they increase at the same rate. So, \(\lim _{x \rightarrow \infty} \frac{p(x)}{q(x)} = \frac{a_{n}}{b_{m}}\), which is the ratio of the leading coefficients.

Case 3: Degree of \(p(x)\) is larger than degree of \(q(x)\)

If \(n>m\), then the degree of \(p(x)\) is higher than that of \(q(x)\). As \(x\) approaches infinity, \(p(x)\) will thus increase faster than \(q(x)\). Therefore, depending whether \(p(x)\) and \(q(x)\) are positive or negative for \(x\rightarrow \infty\), \(\lim_{x \rightarrow \infty} \frac{p(x)}{q(x)}\) will approach either \(+\infty\) or \(-\infty\). The sign of the result will be determined by the signs of the leading coefficients \(a_{n}\) and \(b_{m}\).

Key concepts

Limits of Polynomials

When studying the behavior of polynomial functions as the input variable grows without bound, one fundamental concept to grasp is the notion of limits. In simple terms, the limit describes the value that a function approaches as the variable gets closer to a specific point, which in the case of polynomial long-term behavior, is often infinity.

Specifically, for the given exercise where you have two polynomials, one represented as p(x) and the other as q(x), the limit as x approaches infinity of the ratio p(x)/q(x) can have different outcomes based on their degrees and leading coefficients. If the degree of p(x) is less than the degree of q(x) (when n < m), the function p(x) grows slower than q(x) as x gets very large. Thus, the fraction p(x)/q(x) gets closer and closer to zero. On the other hand, if p(x) has a larger degree compared to q(x) (when n > m), p(x)/q(x) will tend to infinity since p(x) grows much more rapidly. When both polynomials have the same degree, the ratio of their leading coefficients becomes the limit.

Polynomial Degree Comparison

Understanding how the degree of a polynomial affects its growth is crucial in analyzing the limits of polynomials. The degree of a polynomial is the highest power of x that appears with a non-zero coefficient. It determines how rapidly the polynomial's value changes as x increases.

In the context of our problem, comparing the degrees of p(x) and q(x) helps us predict the behavior of their ratio as x tends to infinity.

Paying Attention to Degrees in Limits

When the degree of p(x) is lower than that of q(x) (n < m), the highest power of x will govern the growth of q(x) and therefore, the division p(x)/q(x) shrinks closer to zero as x extends to infinity. Conversely, if p(x) has a higher degree than q(x), it outgrows q(x) leading to the ratio widening towards infinity.

Leading Coefficient Ratio

The leading coefficient of a polynomial is the coefficient of the term with the highest power of x. It plays a prominent role in determining the end behavior of a polynomial function. When two polynomials have the same degree and we are interested in their behavior at infinity, the leading coefficient ratio becomes the deciding factor.

In our exercise where p(x) and q(x) both have the same degree n=m, the limit of their ratio is the ratio of their leading coefficients: a_n over b_m. This happens because as x becomes very large, the terms containing lower powers of x have a negligible impact compared to the leading term.

Importance of Leading Coefficients in Equal Degree Polynomials

Thus, for polynomials of the same degree, the leading coefficient ratio a_n/b_m indicates the factor by which the function's outputs will scale in relation to one another for large x values. If the degrees are unequal, this ratio becomes irrelevant in the limit, since the higher degree term dictates the polynomial's growth rate.