?If a function f represents a system that varies in time, the existence of \(\lim

Explain the meaning of \(\lim \limits_{x \rightarrow-\infty} f(x)=10\).

What is a horizontal asymptote?

Determine \(\lim \limits_{x \rightarrow \infty} \frac{f(x)}{g(x)}\) if \(f(x) \rightarrow 100,000\) and \(g(x) \rightarrow \infty\) as \(x \rightarrow \infty\).

Describe the end behavior of \(g(x)=e^{-2 x}\).

Describe the end behavior of \(f(x)=-2 x^{3}\).

Three cases arise when examining the end behavior of a rational function f(x) = p(x) / g(x). State each case, and describe the associated end behavior.

Evaluate \(\lim \limits_{x \rightarrow \infty} e^{x}, \lim \limits_{x \rightarrow-\infty} e^{x}\), and \(\lim \limits_{x \rightarrow \infty} e^{-x}\).

Describe with a sketch the end behavior of f(x) = In x.

Evaluate the following limits. \(\lim \limits_{x \rightarrow \infty}\left(3+\frac{10}{x^{2}}\right)\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow \infty}\left(5+\frac{1}{x}+\frac{10}{x^{2}}\right)\)

Evaluate the following limits. \(\lim _{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^{2}}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow \infty} \frac{3+2 x+4 x^{2}}{x^{2}}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow-\infty} \frac{\cos x^{5}}{x}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow-\infty}\left(5+\frac{100}{x}+\frac{\sin ^{4} x^{3}}{x^{2}}\right)\)

Determine the following limits. \(\lim \limits_{x \rightarrow \infty}\left(3 x^{12}-9 x^{7}\right)\)

Determine the following limits. \(\lim \limits_{x \rightarrow-\infty}\left(3 x^{7}+x^{2}\right)\)

Determine the following limits. \(\lim \limits_{x \rightarrow-\infty}\left(-3 x^{16}+2\right)\)

Determine the following limits. \(\lim \limits_{x \rightarrow-\infty} 2 x^{-8}\)

Determine the following limits. \(\lim \limits_{x \rightarrow \infty}\left(-12 x^{-5}\right)\)

Determine the following limits. \(\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right)\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of f (if any). \(f(x)=\frac{6 x^{2}-9 x+8}{3 x^{2}+2}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of f (if any). \(f(x)=\frac{4 x^{2}-7}{8 x^{2}+5 x+2}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of f (if any). \(f(x)=\frac{2 x+1}{3 x^{4}-2}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of f (if any). \(f(x)=\frac{12 x^{8}-3}{3 x^{8}-2 x^{7}}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of f (if any). \(f(x)=\frac{40 x^{5}+x^{2}}{16 x^{4}-2 x}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of f (if any). \(f(x)=\frac{-x^{3}+1}{2 x+8}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptote(s) of f (if any). \(f(x)=\frac{4 x^{3}+1}{2 x^{3}+\sqrt{16 x^{6}+1}}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptote(s) of f (if any). \(f(x)=\frac{4 x^{3}}{2 x^{3}+\sqrt{9 x^{6}+15 x^{4}}}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptote(s) of f (if any). \(f(x)=\frac{\sqrt[3]{x^{6}+8}}{4 x^{2}+\sqrt{3 x^{4}+1}}\)

Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptote(s) of f (if any). \(f(x)=4 x\left(3 x-\sqrt{9 x^{2}+1}\right)\)

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. \(f(x)=-3 e^{-x}\)

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. \(f(x)=2^{x}\)

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. f(x) = 1 - ln x

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. f(x) = |ln x|

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. f(x) = sin x

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. \(f(x)=\frac{50}{e^{2 x}}\)

Determine whether the following statements are true and give an explanation or counterexample. a. The graph of a function can never cross one of its horizontal asymptotes. b. A rational function f can have both \(\lim \limits_{x \rightarrow \infty} f(x)=L\) and \(\lim \limits_{x \rightarrow-\infty} f(x)=\infty\). c. The graph of any function can have at most two horizontal asymptotes.

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{x^{2}-4 x+3}{x-1}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{2 x^{3}+10 x^{2}+12 x}{x^{3}+2 x^{2}}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{\sqrt{16 x^{4}+64 x^{2}}+x^{2}}{2 x^{2}-4}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{3 x^{4}+3 x^{3}-36 x^{2}}{x^{4}-25 x^{2}+144}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=16 x^{2}\left(4 x^{2}-\sqrt{16 x^{4}+1}\right)\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{x^{2}-9}{x(x-3)}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{x-1}{x^{2 / 3}-1}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{\sqrt{x^{2}+2 x+6}-3}{x-1}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\frac{\left|1-x^{2}\right|}{x(x+1)}\)

a. Evaluate \(\lim \limits_{x \rightarrow \infty} f(x)\) and \(\lim \limits_{x \rightarrow-\infty} f(x)\), and then identify the horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, evaluate \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\). \(f(x)=\sqrt{|x|}-\sqrt{|x-1|}\)

The central branch of f(x) = tan x is shown in the figure. a. Evaluate \(\lim \limits_{x \rightarrow \pi / 2^{-}} \tan x\) and \(\lim \limits_{x \rightarrow-\pi / 2^{+}} \tan x\). Are these limits infinite limits or limits at infinity? b. Sketch a graph of \(g(x)=\tan ^{-1} x\) by reflecting the graph of f over the line y = x, and use it to evaluate \(\lim \limits_{x \rightarrow \infty} \tan ^{-1} x\) and \(\lim \limits_{x \rightarrow-\infty} \tan ^{-1} x\).

Graph \(y=\sec ^{-1} x\) and evaluate the following limits using the graph. Use the domain given in the text. a. \(\lim \limits_{x \rightarrow \infty} \sec ^{-1} x\) b. \(\lim \limits_{x \rightarrow-\infty} \sec ^{-1} x\)

The hyperbolic cosine function, denoted cosh x, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as \(\cosh x=\frac{e^{x}+e^{-x}}{2}\). a. Determine its end behavior by evaluating \(\lim \limits_{x \rightarrow \infty} \cosh x\) and \(\lim \limits_{x \rightarrow-\infty} \cosh x\). b. Evaluate cosh 0. Use symmetry and part (a) to sketch a plausible graph for y = cosh x.

The hyperbolic sine function is defined as \(\sinh x=\frac{e^{x}-e^{-x}}{2}\). a. Determine its end behavior by evaluating \(\lim \limits_{x \rightarrow \infty} \sinh x\) and \(\lim \limits_{x \rightarrow-\infty} \sinh x\). b. Evaluate sinh 0. Use symmetry and part (a) to sketch a plausible graph for y = sinh x.

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to sketch all vertical and horizontal asymptotes. \(\begin{array}{l} f(-1)=-2, f(1)=2, f(0)=0, \lim \limits_{x \rightarrow \infty} f(x)=1, \\ \lim \limits_{x \rightarrow-\infty} f(x)=-1 \end{array}\)

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to sketch all vertical and horizontal asymptotes. \(\begin{array}{l} \lim \limits_{x \rightarrow 0^{+}} f(x)=\infty, \lim \limits_{x \rightarrow 0^{-}} f(x)=-\infty, \lim \limits_{x \rightarrow \infty} f(x)=1, \\ \lim \limits_{x \rightarrow-\infty} f(x)=-2 \end{array}\)

Find the vertical and horizontal asymptotes of \(f(x)=e^{1 / x}\).

Find the vertical and horizontal asymptotes of \(f(x)=\frac{\cos x+2 \sqrt{x}}{\sqrt{x}}\).

If a function f represents a system that varies in time, the existence of \(\lim \limits_{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state-exists and give the steady-state value. The population of a bacteria culture is given by \(p(t)=\frac{2500}{t+1}\).

If a function f represents a system that varies in time, the existence of \(\lim \limits_{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state-exists and give the steady-state value. The population of a culture of tumor cells is given by \(p(t)=\frac{3500 t}{t+1}\).

If a function f represents a system that varies in time, the existence of \(\lim \limits_{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state-exists and give the steady-state value. The amount of drug (in mg) in the blood after an IV tube is inserted is \(m(t)=200\left(1-2^{-t}\right)\).

If a function f represents a system that varies in time, the existence of \(\lim \limits_{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state-exists and give the steady-state value. The value of an investment is given by \(v(t)=\$ 1000 e^{0.065 t}\).

If a function f represents a system that varies in time, the existence of \(\lim \limits_{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state-exists and give the steady-state value. The population of a colony of squirrels is given by \(p(t)=\frac{1500}{3+2 e^{-0.1 t}}\).

If a function f represents a system that varies in time, the existence of \(\lim \limits_{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state-exists and give the steady-state value. The amplitude of an oscillator is given by \(a(t)=2\left(\frac{t+\sin t}{t}\right)\).

A sequence is an infinite ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,... } is specified by the function f(n) = 2n, where n = 1, 2, 3,.... The limit of such a sequence is \(\lim \limits_{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\{4,2, \frac{4}{3}, 1, \frac{4}{5}, \frac{2}{3}, \ldots\right\}\), which is defined by \(f(n)=\frac{4}{n}\) for n = 1, 2, 3, . . .

A sequence is an infinite ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,... } is specified by the function f(n) = 2n, where n = 1, 2, 3,.... The limit of such a sequence is \(\lim \limits_{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\right\}\), which is defined by \(f(n)=\frac{n-1}{n}\) for n = 1, 2, 3, . . .

A sequence is an infinite ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,... } is specified by the function f(n) = 2n, where n = 1, 2, 3,.... The limit of such a sequence is \(\lim \limits_{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\{\frac{1}{2}, \frac{4}{3}, \frac{9}{4}, \frac{16}{5}, \ldots\right\}\), which is defined by \(f(n)=\frac{n^{2}}{n+1}\) for n = 1, 2, 3, . . .

A sequence is an infinite ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,... } is specified by the function f(n) = 2n, where n = 1, 2, 3,.... The limit of such a sequence is \(\lim \limits_{n \rightarrow \infty} f(n)\), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences, or state that the limit does not exist. \(\left\{2, \frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \ldots\right\}\), which is defined by \(f(n)=\frac{n+1}{n^{2}}\) for n = 1, 2, 3, . . .

Suppose p/q is a rational function where the degree of p is 1 greater than the degree of q. Using polynomial long division, p/q can be written as \(\frac{p(x)}{q(x)}=m x+b+\frac{r(x)}{s(x)}\) where r/s is a rational function with the property \(r(x) / s(x) \rightarrow 0\) as \(x \rightarrow \pm \infty\). This fact implies that \(\frac{p(x)}{q(x)} \approx m x+b\) when x is large. The line y = mx + b is an oblique (or slant) asymptote of p/q. Complete the following steps for the given functions. a. Use polynomial long division to find the oblique asymptote of f. b. Find the vertical asymptotes of f. c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. \(f(x)=\frac{x^{2}-1}{x+2}\)

Suppose p/q is a rational function where the degree of p is 1 greater than the degree of q. Using polynomial long division, p/q can be written as \(\frac{p(x)}{q(x)}=m x+b+\frac{r(x)}{s(x)}\) where r/s is a rational function with the property \(r(x) / s(x) \rightarrow 0\) as \(x \rightarrow \pm \infty\). This fact implies that \(\frac{p(x)}{q(x)} \approx m x+b\) when x is large. The line y = mx + b is an oblique (or slant) asymptote of p/q. Complete the following steps for the given functions. a. Use polynomial long division to find the oblique asymptote of f. b. Find the vertical asymptotes of f. c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. \(f(x)=\frac{x^{2}-3}{x+6}\)

Suppose p/q is a rational function where the degree of p is 1 greater than the degree of q. Using polynomial long division, p/q can be written as \(\frac{p(x)}{q(x)}=m x+b+\frac{r(x)}{s(x)}\) where r/s is a rational function with the property \(r(x) / s(x) \rightarrow 0\) as \(x \rightarrow \pm \infty\). This fact implies that \(\frac{p(x)}{q(x)} \approx m x+b\) when x is large. The line y = mx + b is an oblique (or slant) asymptote of p/q. Complete the following steps for the given functions. a. Use polynomial long division to find the oblique asymptote of f. b. Find the vertical asymptotes of f. c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. \(f(x)=\frac{3 x^{2}-2 x+7}{2 x-5}\)

Suppose p/q is a rational function where the degree of p is 1 greater than the degree of q. Using polynomial long division, p/q can be written as \(\frac{p(x)}{q(x)}=m x+b+\frac{r(x)}{s(x)}\) where r/s is a rational function with the property \(r(x) / s(x) \rightarrow 0\) as \(x \rightarrow \pm \infty\). This fact implies that \(\frac{p(x)}{q(x)} \approx m x+b\) when x is large. The line y = mx + b is an oblique (or slant) asymptote of p/q. Complete the following steps for the given functions. a. Use polynomial long division to find the oblique asymptote of f. b. Find the vertical asymptotes of f. c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. \(f(x)=\frac{x^{2}-2 x+5}{3 x-2}\)

Suppose p/q is a rational function where the degree of p is 1 greater than the degree of q. Using polynomial long division, p/q can be written as \(\frac{p(x)}{q(x)}=m x+b+\frac{r(x)}{s(x)}\) where r/s is a rational function with the property \(r(x) / s(x) \rightarrow 0\) as \(x \rightarrow \pm \infty\). This fact implies that \(\frac{p(x)}{q(x)} \approx m x+b\) when x is large. The line y = mx + b is an oblique (or slant) asymptote of p/q. Complete the following steps for the given functions. a. Use polynomial long division to find the oblique asymptote of f. b. Find the vertical asymptotes of f. c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. \(f(x)=\frac{3 x^{2}-2 x+5}{3 x+4}\)

Suppose p/q is a rational function where the degree of p is 1 greater than the degree of q. Using polynomial long division, p/q can be written as \(\frac{p(x)}{q(x)}=m x+b+\frac{r(x)}{s(x)}\) where r/s is a rational function with the property \(r(x) / s(x) \rightarrow 0\) as \(x \rightarrow \pm \infty\). This fact implies that \(\frac{p(x)}{q(x)} \approx m x+b\) when x is large. The line y = mx + b is an oblique (or slant) asymptote of p/q. Complete the following steps for the given functions. a. Use polynomial long division to find the oblique asymptote of f. b. Find the vertical asymptotes of f. c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. \(f(x)=\frac{4 x^{3}+4 x^{2}+7 x+4}{1+x^{2}}\)

Suppose \(f(x)=\frac{p(x)}{q(x)}\) is a rational function, where \(\begin{array}{l} p(x)=a_{m} x^{m}+a_{m-1} x^{m-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}, \\ q(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{2} x^{2}+b_{1} x+b_{0}, a_{m} \neq 0, \\ \text { and } b_{n} \neq 0. \end{array}\) a. Prove that if m = n, then \(\lim \limits_{x \rightarrow \pm \infty} f(x)=\frac{a_{m}}{b_{n}}\). b. Prove that if m < n, then \(\lim \limits_{x \rightarrow \pm \infty} f(x)=0\).