Chapter 3

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{x}{1-e^{-x}} $$

(a) Use a graphing calculator to sketch the graph of $$f(x)=e^{a x} \sin x, \quad x \geq 0$$ for \(a=-0.1,-0.01,0,0.01\), and 0.1. (b) Which part of the function \(f(x)\) produces the oscillations that you see in the graphs sketched in (a)? (c) Describe in words the effect that the value of \(a\) has on the shape of the graph of \(f(x)\). (d) Graph \(f(x)=e^{a x} \sin x, g(x)=-e^{a x}\), and \(h(x)=e^{a x}\) together in one coordinate system for (i) \(a=0.1\) and (ii) \(a=-0.1\). [Make separate graphs for (i) and (ii).] Explain what you see in each case. Show that $$-e^{a x} \leq e^{a x} \sin x \leq e^{a x}$$ Use this pair of inequalities to determine the values of \(a\) for which \(\lim _{x \rightarrow \infty} f(x)\) exists, and find the limiting value.

Use the formal definition of limits to prove each statement. \(\lim _{x \rightarrow c}(m x)=m c\), where \(m\) is a constant

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\frac{x}{x+1} $$

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \frac{3}{2+e^{-x}} $$

Use the formal definition of limits to prove each statement. \(\lim _{x \rightarrow c}(m x+b)=m c+b\), where \(m\) and \(b\) are constants

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\exp [\sqrt{x-1}] $$

Evaluate the limits. $$ \lim _{x \rightarrow-\infty} \frac{4}{1+e^{-x}} $$

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\tan (2 \pi x) $$

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \frac{2}{e^{x}(1+x)} $$

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 1^{-}} \frac{2}{1-x} $$

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\cos \left(\frac{2 x}{3+x}\right) $$

Evaluate the limits. $$ \lim _{x \rightarrow-\infty} \frac{e^{x}}{1+x} $$

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 2^{+}} \frac{4}{2-x} $$

Let $$ f(x)=\left\\{\begin{array}{ll} x^{2}+2 & \text { for } x \leq 0 \\ x+c & \text { for } x>0 \end{array}\right. $$ (a) Graph \(f(x)\) when \(c=1\), and determine whether \(f(x)\) is continuous for this choice of \(c\). (b) How must you choose \(c\) so that \(f(x)\) is continuous for all \(x \in(-\infty, \infty)\) ?

The Monod model is used to describe how the rate of reproduction of organisms depends on the amount of nutrients that are available. Monod (1949) studied how the rate of division of \(E\). coli cells depended upon the amount of sugar added to their growth flask. If \(r\) is the rate of reproduction (number of divisions in one hour) and \(C\) is the amount of glucose sugar added to the growth medium measured in moles then: $$r(C)=\frac{1.35 C}{C+0.22 \times 10^{-4}}$$ (a) Show that the reproduction rate goes to zero when the sugar level is low; that is: $$\lim _{C \rightarrow 0} r(C)=0$$ (b) Show that if more and more sugar is added, the reproductive rate plateaus; that is, \(\lim _{C \rightarrow \infty} r(C)\) exists and calculate this limit.

Let $$ f(x)=\left\\{\begin{array}{cl} \frac{1}{x} & \text { for } x \geq 1 \\ 2 x+c & \text { for } x<1 \end{array}\right. $$ (a) Graph \(f(x)\) when \(c=0\), and determine whether \(f(x)\) is continuous for this choice of \(c\). \begin{array}{l} \text { (b) How must you choose } c \text { so that } f(x) \text { is continuous for all }\\\ x \in(-\infty, \infty) ? \end{array}

The Hill equation is used to model how hemoglobin in blood binds to oxygen. If the proportion of hemoglobin molecules that are bound to oxygen is \(h\) and the concentration of oxygen (measured as a partial pressure, that varies from 0 to \(\infty\) ) is \(P\), then a common model is: $$h(P)=\frac{a P^{k}}{30^{k}+P^{k}}$$ where \(k \geq 1\) and \(a>0\) are constants that depend on the species of animal and its environment (e.g., whether it lives at sea-level or at altitude). (a) Show that no matter what the values of \(a\) and \(k\) are, the amount of bound oxygen goes to zero as the oxygen concentration goes to \(0 ;\) that is: $$\lim _{P \rightarrow 0} h(P)=0$$ (b) It is known that as \(P\) increases, the amount of bound oxygen plateaus. Since \(h=1\) when all hemoglobin molecules are bound to oxygen, we want our model to reflect that: $$\lim _{P \rightarrow \infty} h(P)=1$$ This is called the saturation value for oxygen binding. Explain what value of \(a\) must be chosen for this condition to be satisfied. (c) The half-saturation constant, \(P_{1 / 2}\), is defined to be the concentration of oxygen at which the proportion of bound hemoglobin molecules reaches half its saturation value, that is: $$h\left(P_{1 / 2}\right)=\frac{1}{2} \lim _{P \rightarrow \infty} h(P)$$ Show that \(P_{1 / 2}=30\). (d) In a patient with carbon monoxide poisoning carbon monoxide binds preferentially to the hemoglobin instead of oxygen, stopping the blood from effectively transporting oxygen around the body. For a patient with acute carbon monoxide poisoning, the relationship between proportion of bound hemoglobin molecules and oxygen concentration can be modeled by: \(h(P)=\frac{0.9 P^{3}}{P^{3}+26^{3}} \quad\) (we have assumed that \(k=3\) ) Show that both the saturation level for oxygen binding and the half-saturation constant are both changed from your answers in (b) and (c).

(a) Show that $$ f(x)=\sqrt{x-1}, \quad x \geq 1 $$ is continuous from the right at \(x=1\). (b) Graph \(f(x)\). (c) Does it make sense to look at continuity from the left at \(x=1 ?\)

Suppose the size of a population at time \(t\) is given by $$N(t)=\frac{500 t}{3+t}, \quad t \geq 0.$$ (a) Use a graphing calculator to sketch the graph of \(N(t)\). (b) Determine the size of the population as \(t \rightarrow \infty\). We call this the limiting population size. (c) Show that, at time \(t=3\), the size of the population is half its limiting size.

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 3} \frac{1}{(x-3)^{2}} $$

(a) Show that $$ f(x)=\sqrt{x^{2}-4}, \quad|x| \geq 2 $$ is continuous from the right at \(x=2\) and continuous from the left at \(x=-2\). (b) Graph \(f(x)\). (c) Does it make sense to look at continuity from the left at \(x=2\) and at continuity from the right at \(x=-2 ?\)

Suppose that the size of a population at time \(t\) is given by $$N(t)=\frac{50}{1+6 e^{-2 t}}, \quad t \geq 0$$ (a) Use a graphing calculator to sketch the graph of \(N(t)\). (b) Determine the size of the population as \(t \rightarrow \infty\), using the basic rules for limits. Compare your answer with the graph that you sketched in (a).

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{1-x^{2}}{x^{2}} $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow \pi / 3} \sin \left(\frac{x}{2}\right) $$

Suppose that the size of a population at time \(t\) is given by $$N(t)=\frac{100}{1+3 e^{-t}}, \quad t \geq 0$$ (a) Use a graphing calculator to sketch the graph of \(N(t)\). (b) Determine the size of the population as \(t \rightarrow \infty\), using the basic rules for limits. Compare your answer with the graph that you sketched in (a).

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0^{+}} x \ln x $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-\pi / 2} \cos (2 x) $$

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0^{+}} x^{2} \ln x $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow \pi / 2} \frac{\cos ^{2} x}{1-\sin ^{2} x} $$

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x^{2}} $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-\pi / 2} \frac{1+\tan ^{2} x}{\sec ^{2} x} $$

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x} $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-1} \sqrt{4+5 x^{4}} $$

Use a table and a graph to find out what happens to $$ f(x)=\frac{2}{x}-\frac{1}{x^{2}} $$ as \(x \rightarrow 0^{-}\). What happens as \(x \rightarrow 0^{+}\) ?

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-2} \sqrt{6+x} $$

Use a table and a graph to find out what happens to $$ f(x)=\exp \left(\frac{1}{x}\right) $$ as \(x \rightarrow 0\)

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-1} \sqrt{x^{2}+2 x+2} $$

Use a graphing calculator to investigate $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 1} \sqrt{x^{3}+4 x-1} $$

Use a graphing calculator to investigate $$ \lim _{x \rightarrow 0^{+}}\left(\frac{1}{x^{1 / 2}}-\frac{1}{x}\right) $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} e^{-x^{2} / 3} $$

Tumor Size The Gompertz function is used to model the growth of tumors with time. According to the Gompertz function, the number of cells in a tumor increases with time according to: $$ N(t)=A \exp \left(-b e^{-c t}\right) $$ where \(A, b\), and \(c\) are all positive constants that take different values for different tumor types and depending on whether the tumor is being treated or not. (a) Assume that \(A=b=c=1\). Use a table or a graph to calculate \(\lim _{t \rightarrow 0} N(t)\). (b) Can you explain (without evaluating \(N(t)\) ) why doubling \(A\) will double \(\lim _{t \rightarrow 0} N(t) ?\) (c) Show (it is okay to calculate \(N(t))\) that changing the value of \(b\) changes \(\lim _{t \rightarrow 0} N(t)\) but changing \(c\) does not affect \(\lim _{t \rightarrow 0} N(t)\).

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} e^{3 x+2} $$

Rate of Growth of a Tumor The rate of proliferation (that is, reproduction) for the cells in a tumor varies depending on the size of the tumor. The Gompertz growth model is sometimes used to model this growth. According to the Gompertz model the total number of divisions occurring in 1 hour, \(R\), depends on the number of cells, \(N\), through a formula: $$ R(N)=d N \ln \left(\frac{A}{N}\right) $$ where \(d\) and \(A\) are both positive constants that depend on the type of tumor, whether it is being treated or not, and so on. (a) Assume that \(d=A=1\). Use a table or a graph to show that \(\lim _{N \rightarrow 0} R(N)=0\) (b) The per cell rate of reproduction tells us how many times any cell in the tumor will divide in one hour. It is given by \(r(N)=\frac{R(N)}{N} .\) Show that \(\lim _{N \rightarrow 0} r(N)\) does not exist (again assume that \(d=A=1\) ).

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 3} e^{x^{2}-9} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-1}\left(x^{3}+7 x-1\right) $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-1} e^{x^{2} / 2-1} $$

In Problems 39-56, use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 2}\left(3 x^{4}-2 x+1\right) $$

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 0} \frac{e^{2 x}-1}{e^{x}-1} $$