Chapter 1.3, Problem 25E

Solution

To Find: The suitable functions from twelve basic functions based on the given description.

The four functions that do not have end behaviour $\underset{x\to +\infty }{\lim }f\left(x\right)=+\infty$ are:

  $\begin{array}{c}y=\frac{1}{x}\\ y=\sin x\\ y=\cos x\\ y=\frac{1}{1+e^{-x}}\end{array}$

Given information:

The four functions that do not have end behaviour $\underset{x\to +\infty }{\lim }f\left(x\right)=+\infty$

Given functions are:

  $\begin{array}{l}1.f\left(x\right)=x2.f\left(x\right)=x^2\\ 3.f\left(x\right)=x^{3}4.f\left(x\right)=\frac{1}{x}\\ 5.f\left(x\right)=\sqrt{x}6.f\left(x\right)=e^x\\ 7.f\left(x\right)=\ln x8.f\left(x\right)=\sin x\\ 9.f\left(x\right)=\cos x10.f\left(x\right)=\left|x\right|\\ 11.f\left(x\right)=\mathrm{int}x12.f\left(x\right)=\frac{1}{1+e^{-x}}\end{array}$

Calculation:

As a function goes off the $x-$ axis towards either end, the horizontal asymptote gives one kind of end behaviour.

The four functions that do not have end behaviour are:

  $\begin{array}{c}y=\frac{1}{x}\\ y=\sin x\\ y=\cos x\\ y=\frac{1}{1+e^{-x}}\end{array}$

The graphical representation of the functions $y=\frac{1}{x},y=\sin x,y=\cos x,y=\frac{1}{1+e^{-x}}$ is:

  

From the figure, it is clear that the graphs of $y=\frac{1}{x},y=\sin x,y=\cos x$ and $y=\frac{1}{1+e^{-x}}$ do not have end behaviour since it does not satisfies the ideology of end behaviour and also when $\underset{x\to +\infty }{\lim }f\left(x\right)=+\infty$ .

The functions become undefined when substitute $x=\infty$ and there is no horizontal asymptote.

Thus, the four functions that do not have end behaviour $\underset{x\to +\infty }{\lim }f\left(x\right)=+\infty$ are:

  $\begin{array}{c}y=\frac{1}{x}\\ y=\sin x\\ y=\cos x\\ y=\frac{1}{1+e^{-x}}\end{array}$