In each part, find examples of polynomials p(x) and q(x)...

In each part, find examples of polynomials $p(x)$ and $q(x)$ that satisfy the stated condition and such that $p(x) \rightarrow+\infty$ and $q(x) \rightarrow+\infty$ as $x \rightarrow+\infty$. $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow+\infty} \frac{p(x)}{q(x)}=1} & {\text { (b) } \lim _{x \rightarrow+\infty} \frac{p(x)}{q(x)}=0} \\ {\text { (c) } \lim _{x \rightarrow+\infty} \frac{p(x)}{q(x)}=+\infty} & {\text { (d) } \lim _{x \rightarrow+\infty}[p(x)-q(x)]=3}\end{array} $$

In each part, find examples of polynomials $p(x)$ and $q(x)$
that satisfy the stated condition and such that $p(x) \rightarrow+\infty$
and $q(x) \rightarrow+\infty$ as $x \rightarrow+\infty$.
$$
\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow+\infty} \frac{p(x)}{q(x)}=1} & {\text { (b) } \lim _{x \rightarrow+\infty} \frac{p(x)}{q(x)}=0} \\ {\text { (c) } \lim _{x \rightarrow+\infty} \frac{p(x)}{q(x)}=+\infty} & {\text { (d) } \lim _{x \rightarrow+\infty}[p(x)-q(x)]=3}\end{array}
$$

Key Concepts

Polynomial Degree Comparison

The behavior of rational functions as x approaches infinity is determined primarily by the degrees of the polynomials in the numerator and denominator. If both polynomials have the same degree, the dominant terms cancel out leaving a finite limit equal to the ratio of the leading coefficients. When the degree of the polynomial in the numerator is less than that in the denominator, the ratio approaches zero; conversely, if it is greater, the ratio grows without bound.

Leading Coefficient

For polynomials of equal degree, the limit of their ratio as x approaches infinity depends on the leading coefficients. The leading coefficient is the coefficient of the term with the highest degree and determines the main contribution to the value of the polynomial for large values of x, making it crucial for determining the end behavior of the function.

Limit of Functions

The concept of a limit is used to describe the behavior of a function as the input approaches a particular value, often infinity. It provides a formal way of understanding the end behavior of functions, particularly in cases where the function tends to a constant, zero, or infinity.

Asymptotic Behavior

Asymptotic behavior refers to the behavior of functions as the variable approaches a certain limit, typically infinity. For polynomials and rational functions, this concept is utilized to analyze how the dominant, highest power terms govern the growth or decay of the function, thereby simplifying the evaluation of limits.

Construction of Specific Functions

When constructing examples of functions that meet certain limiting criteria, the degrees of the polynomials and the values of their leading coefficients need to be tailored to produce the desired behavior. This involves deliberate manipulation of the polynomial terms so that as x increases indefinitely, the functions exhibit the required limit (equal to a constant, zero, or infinity) or a specific difference.

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