Chapter 3

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=\ln (x-2) $$

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

(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) \(a=\) \(-0.1 .\) [Use separate coordinate systems 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 \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=\ln \frac{x}{x+1} $$

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

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 \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=\exp [-\sqrt{x-1}] $$

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

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

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

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

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

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

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 \(x \in(-\infty, \infty) ?\)

In Section 1.2.3, Example 6, we introduced the Monod growth function $$r(N)=a \frac{N}{k+N}, \quad N \geq 0$$ Find \(\lim _{N \rightarrow \infty} r(N)\).

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

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 \(c\). (b) How must you choose \(c\) so that \(f(x)\) is continuous for all \(x \in(-\infty, \infty)\) ?

In Problem 86 of Section \(1.3\), we discussed the MichaelisMenten equation, which describes the initial velocity of an enzymatic reaction \(\left(v_{0}\right)\) as a function of the substrate concentration \(\left(s_{0}\right)\). The equation was given by $$v_{0}=\frac{v_{\max } s_{0}}{s_{0}+K_{m}}$$ Find \(\lim _{s_{0} \rightarrow \infty} v_{0}\).

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 $$ f(x)=\sqrt{x-1}, \quad x \geq 1 $$ 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. $$ \lim _{x \rightarrow 0} \frac{1-x^{2}}{x^{2}} $$

(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 risht at \(x-\infty\)

Suppose that the size of a population at time \(t\) is given by $$N(t)=\frac{100}{1+9 e^{-t}}$$ for \(t \geq 0\). (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. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+9}-3}{x^{2}} $$

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 $$N(t)=\frac{50}{1+3 e^{-t}}$$ for \(t \geq 0\). (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. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x^{2}+4}-2}{x} $$

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. $$ \lim _{x \rightarrow 0} \frac{1-\sqrt{1-x^{2}}}{x^{2}} $$

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. $$ \lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2}}{2 x} $$

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

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 $$ f(x)=\frac{2 x}{x-1} $$ as \(x \rightarrow \infty\). What happens as \(x \rightarrow-\infty\) ? What happens as \(x \rightarrow 1 ?\)

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

Use a graphing calculator to investigate $$ \lim _{x \rightarrow 1} \sin \frac{1}{x-1} $$

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

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

In Problems \(37-54\), 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. $$ \lim _{x \rightarrow 0} e^{-x^{2} / 3} $$

In Problems \(37-54\), 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. $$ \lim _{x \rightarrow 0} e^{3 x+2} $$

In Problems \(37-54\), use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-5}\left(4+2 x^{2}\right) $$

In Problems \(37-54\), use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 2}\left(8 x^{3}-2 x+4\right) $$

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

In Problems \(37-54\), use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow 3}\left(2 x^{2}-\frac{1}{x}\right) $$

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

In Problems \(37-54\), use the limit laws to evaluate each limit. $$ \lim _{x \rightarrow-2}\left(\frac{x^{2}}{2}-\frac{2}{x^{2}}\right) $$