Problem 101 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

The conclusion is that the limit of the rational function depends on the relative degrees of the numerator and the denominator polynomials. When \(n < m\), the limit is 0; when \(n = m\), the limit is \(\frac{a_{n}}{b_{m}}\); and when \(n > m\), the limit is ± infinity.

Step 1: Case 1: n < m

Assuming \(n < m\), we can rearrange the expression into this form: \[ \frac{p(x)}{q(x)} = \frac{a_{n} x^{n}+\cdots+a_{1} x+a_{0}}{b_{m} x^{m}+\cdots+b_{1} x+b_{0}} = \frac{a_{n}/x^{m-n} + \cdots + a_{1}/x^{m-1} + a_{0}/x^{m}}{b_{m} + \cdots + b_{1}/x^{m-1} + b_{0}/x^{m}} \] As \(x\) approaches infinity, all terms with \(x\) in the denominator will approach 0. Hence, the numerator becomes 0 and the denominator becomes \(b_{m}\). Thus, the limit is 0.

Step 2: Case 2: n = m

If \(n = m\), then the degree of the numerator equals the degree of the denominator. We can then reduce the expression to: \[ \frac{a_{n} x^{n}+\cdots+a_{1} x+a_{0}}{b_{m} x^{m}+\cdots+b_{1} x+b_{0}} = \frac{a_{n} + \cdots + a_{1}/x^{n-1} + a_{0}/x^{n}}{b_{m} + \cdots + b_{1}/x^{n-1} + b_{0}/x^{n}} \] In this case, only the first terms in the numerator and denominator don't tend to zero as \(x\) approaches infinity. Therefore, this limit is just \(\frac{a_{n}}{b_{m}}\) as stated.

Step 3: Case 3: n > m

For the case where \(n > m\), the degree of the numerator is higher than the degree of the denominator. We can simplify the expression to: \[ \frac{a_{n} x^{n} + \cdots + a_{1} x + a_{0}}{b_{m} x^{m} + \cdots + b_{1} x + b_{0}} = \frac{a_{n} + \cdots + a_{m}/x^{n-m} + \cdots + a_{1}/x^{n-1} + a_{0}/x^{n}}{b_{m}/x^{n-m} + \cdots + b_{1}/x^{n-1} + b_{0}/x^{n}} \] In this case, as \(x\) approaches infinity, the first term in the numerator (which is \(a_n\)) will dominate, and all terms in the denominator will tend to 0. This gives us a fraction with non-zero numerator and zero denominator, i.e. ± infinity, depending on the sign of \(a_n\).

Key Concepts

Limit as x Approaches Infinity

Understanding the behavior of polynomials as x approaches infinity is key to mastering calculus. In this context, the limit of a function as x approaches infinity often indicates the function's end behavior. When we say \( \lim_{x \to \infty} f(x) \), we're interested in the value that f(x) approaches as x gets larger and larger.

For polynomial functions, they become simpler the larger x gets; higher exponents on x dominate the value of the function, and constants and lower exponents fade in significance. This concept is essential when dealing with rational functions, where we often divide one polynomial by another. Thus, by analyzing the highest degree terms (those with the largest exponents) in the numerator and denominator, we can often determine the limit of a rational function as x approaches infinity.

Polynomial Functions

Polynomial functions are foundational in mathematics and appear as expressions like \( p(x) = a_n x^n + \cdots + a_1 x + a_0 \), where n represents the degree of the polynomial and the a's are coefficients. These functions have the nice property that they are continuous and differentiable everywhere, meaning we can graph them without lifting our pencil from the paper, and we can calculate their rates of change at any point.

When evaluating limits, we focus on the coefficient of the highest power of x, as it dominates the end behavior of the polynomial. As such, the degree of the polynomial and the leading coefficient heavily influence the limits as x approaches infinity.

Indeterminate Forms

An indeterminate form arises when the limit of an expression does not lead to a conclusive value without further analysis. Common examples include \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). These expressions do not have well-defined mathematical values, so they require additional techniques, like factoring, conjugate multiplication, or L'Hôpital's rule, to resolve.

In the context of polynomial limits, indeterminate forms most often appear when we're dealing with ratios of polynomials whose degrees are equal, indicated by \( n = m \). In this case, further simplification is key to determining the exact limit value instead of concluding an indeterminate form.

Rational Functions

Rational functions are ratios of polynomials, expressed as \( \frac{p(x)}{q(x)} \) with their own interesting characteristics. They can have vertical asymptotes, horizontal asymptotes, or oblique asymptotes, which describe their behavior as x goes to infinity or approaches certain critical points.

When the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator (n < m), the horizontal asymptote is y = 0, indicating the function approaches zero as x gets large. If the degrees are equal (n = m), the function has a horizontal asymptote at y = a_n/b_m. When the numerator's degree is greater (n > m), the function increases or decreases without bound, and no horizontal asymptote exists. Thus, the degree comparison is a quick tool to determine the limit and behavior of rational functions at infinity.