Chapter 3

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{1-x^{3}}{2+x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0} x^{5}=0 $$

Suppose that the number of individuals in a population at time \(t\) is given by $$ N(t)=\frac{54 t}{13+t}, \quad t \geq 0 $$ (a) Use a calculator to confirm that \(N(10)\) is approximately 23.47826. Considering that the number of individuals in a population is an integer, how should you report your answer? (b) Now suppose that \(N(t)\) is given by the same function (3.7), but that the size of the population is measured in millions. How should you report the population size at time \(t=10 ?\) Make some reasonable assumptions about the accuracy of a measurement for the size of such a large population. (c) Discuss the use of continuous functions in both (a) and (b).

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

In Problems 9-12, determine at which points \(f(x)\) is discontinuous. $$ f(x)=\left\\{\begin{array}{cl} \frac{x^{2}-3 x+2}{x-2} & \text { if } x \neq 1 \\ 1 & \text { if } x=1 \end{array}\right. $$

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

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 1} \frac{1}{x}=1 $$

Suppose that the biomass of a population at time \(t\) is given by $$ B(t)=\frac{32.00 t}{17.00+t}, \quad t \geq 0 $$ (a) Use a calculator to confirm that \(B(10)\) is approximately 1.185185. Considering the function \(B(t)\), how many significant figures should you report in your answer? (b) Discuss the use of continuous functions in this problem.

In Problems 9-12, determine at which points \(f(x)\) is discontinuous. $$ f(x)=\left\\{\begin{array}{cl} x^{2}-1 & \text { if } x \leq 0 \\ x & \text { if } x>0 \end{array}\right. $$

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \frac{2 x+x^{2}}{3 x+1} $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin x \cos x}{x(1-x)} $$

Explain why a polynomial of degree 3 has at least one root.

In Problems 1-32, use a table or a graph to investigate each limit. $$ \lim _{x \rightarrow 3} \frac{x^{2}-16}{x-4} $$

Show that the floor function \(f(x)=\lfloor x\rfloor\) is continuous at \(x=5 / 2\) but discontinuous at \(x=3\).

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \frac{4}{1+e^{-2 x}} $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos ^{2} x}{x^{2}} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0} \frac{-2}{x^{2}}=-\infty $$

Explain why a polynomial of degree \(n\), where \(n\) is an odd number, has at least one root.

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

Show that the floor function \(f(x)=\lfloor x\rfloor\) is continuous from the right at \(x=2\).

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

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos x}{2 x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0} \frac{1}{x^{4}}=\infty $$

Explain why \(y=x^{2}-4\) has at least two roots.

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

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=3 x^{4}-x^{2}+4 $$

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

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos (2 x)}{3 x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 3} \frac{1}{(x-3)^{2}}=\infty $$

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

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=\sqrt{x^{2}-1} $$

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

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos (5 x)}{2 x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow \infty} \frac{2}{x^{2}}=0 $$

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

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=\frac{x^{2}+1}{x-1} $$

Evaluate the limits in problems. $$ \lim _{x \rightarrow-\infty} \exp [x] $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{1-\cos (x / 2)}{x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow \infty} e^{-x}=0 $$

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

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

Evaluate the limits in problems. $$ \lim _{x \rightarrow \infty} \exp [-\ln x] $$

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\sin x(1-\cos x)}{x^{2}} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow \infty} \frac{x}{x+1}=1 $$

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

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

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

Evaluate the trigonometric limits. $$ \lim _{x \rightarrow 0} \frac{\csc x-\cot x}{x \csc x} $$

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow-\infty} \frac{x}{x+1}=1 $$

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