2.3 The limit laws  (Page 6/15)

$\underset{x\to 0}{\text{lim}}\left(4x^2-2x+3\right)$

$\underset{x\to 1}{\text{lim}}\frac{x^3+3x^2+5}{4-7x}$

$\underset{x\to \mathrm{-2}}{\text{lim}}\sqrt{x^2-6x+3}$

$\underset{x\to \mathrm{-1}}{\text{lim}}{\left(9x+1\right)}^2$

$\underset{x\to 7}{\text{lim}}{x}^2$

$\underset{x\to \mathrm{-2}}{\text{lim}}\left(4x^2-1\right)$

$\underset{x\to 0}{\text{lim}}\frac{1}{1+\text{sin}x}$

$\underset{x\to 2}{\text{lim}}{e}^{2x-x^2}$

$\underset{x\to 1}{\text{lim}}\frac{2-7x}{x+6}$

$\underset{x\to 3}{\text{lim}}\text{ln}{e}^{3x}$

$\underset{x\to 4}{\text{lim}}\frac{x^2-16}{x-4}$

$\underset{x\to 2}{\text{lim}}\frac{x-2}{x^2-2x}$

$\underset{x\to 6}{\text{lim}}\frac{3x-18}{2x-12}$

$\underset{x\to 0}{\text{lim}}\frac{{\left(1+h\right)}^2-1}{h}$

$\underset{x\to 9}{\text{lim}}\frac{t-9}{\sqrt{t}-3}$

$\underset{h\to 0}{\text{lim}}\frac{\frac{1}{a+h}-\frac{1}{a}}{h},$ where a is a real-valued constant

$\underset{\theta \to \pi }{\text{lim}}\frac{\text{sin}\theta }{\text{tan}\theta }$

$\underset{x\to 1}{\text{lim}}\frac{x^3-1}{x^2-1}$

$\underset{x\to 1\text{/}2}{\text{lim}}\frac{2x^2+3x-2}{2x-1}$

$\underset{x\to \mathrm{-3}}{\text{lim}}\frac{\sqrt{x+4}-1}{x+3}$

$\underset{x\to {\mathrm{-2}}^{-}}{\text{lim}}\frac{2x^2+7x-4}{x^2+x-2}$

$\underset{x\to {\mathrm{-2}}^{+}}{\text{lim}}\frac{2x^2+7x-4}{x^2+x-2}$

$\underset{x\to 1^{-}}{\text{lim}}\frac{2x^2+7x-4}{x^2+x-2}$

$\underset{x\to 1^{+}}{\text{lim}}\frac{2x^2+7x-4}{x^2+x-2}$

$\underset{x\to 6}{\text{lim}}2f\left(x\right)g\left(x\right)$

$\underset{x\to 6}{\text{lim}}\frac{g\left(x\right)-1}{f\left(x\right)}$

$\underset{x\to 6}{\text{lim}}\left(f\left(x\right)+\frac{1}{3}g\left(x\right)\right)$

$\underset{x\to 6}{\text{lim}}\frac{{\left(h\left(x\right)\right)}^3}{2}$

$\underset{x\to 6}{\text{lim}}\sqrt{g\left(x\right)-f\left(x\right)}$

$\underset{x\to 6}{\text{lim}}x·h\left(x\right)$

$\underset{x\to 6}{\text{lim}}\left[\left(x+1\right)·f\left(x\right)\right]$

$\underset{x\to 6}{\text{lim}}\left(f\left(x\right)·g\left(x\right)-h\left(x\right)\right)$

$f\left(x\right)=\{\begin{array}{ll}{x}^2,& x\le 3\hfill \\ x+4,& x>3\hfill \end{array}$

1. $\underset{x\to 3^{-}}{\text{lim}}f\left(x\right)$
2. $\underset{x\to 3^{+}}{\text{lim}}f\left(x\right)$

$g\left(x\right)=\{\begin{array}{ll}{x}^3-1,& x\le 0\hfill \\ 1,& x>0\hfill \end{array}$

1. $\underset{x\to 0^{-}}{\text{lim}}g\left(x\right)$
2. $\underset{x\to 0^{+}}{\text{lim}}g\left(x\right)$

$h\left(x\right)=\{\begin{array}{ll}{x}^2-2x+1,& x<2\hfill \\ 3-x,& x\ge 2\hfill \end{array}$

1. $\underset{x\to 2^{-}}{\text{lim}}h\left(x\right)$
2. $\underset{x\to 2^{+}}{\text{lim}}h\left(x\right)$

$\underset{x\to {\mathrm{-3}}^{+}}{\text{lim}}\left(f\left(x\right)+g\left(x\right)\right)$

$\underset{x\to {\mathrm{-3}}^{-}}{\text{lim}}\left(f\left(x\right)-3g\left(x\right)\right)$

$\underset{x\to 0}{\text{lim}}\frac{f\left(x\right)g\left(x\right)}{3}$

$\underset{x\to \mathrm{-5}}{\text{lim}}\frac{2+g\left(x\right)}{f\left(x\right)}$

$\underset{x\to 1}{\text{lim}}{\left(f\left(x\right)\right)}^2$

$\underset{x\to 1}{\text{lim}}\sqrt{f\left(x\right)-g\left(x\right)}$

$\underset{x\to \mathrm{-7}}{\text{lim}}\left(x·g\left(x\right)\right)$

$\underset{x\to \mathrm{-9}}{\text{lim}}\left[x·f\left(x\right)+2·g\left(x\right)\right]$

[T] True or False? If $2x-1\le g\left(x\right)\le x^2-2x+3,$ then $\underset{x\to 2}{\text{lim}}g\left(x\right)=0.$

[T] $\underset{\theta \to 0}{\text{lim}}{\theta }^2\text{cos}\left(\frac{1}{\theta }\right)$

$\underset{x\to 0}{\text{lim}}f\left(x\right),$ where $f\left(x\right)=\{\begin{array}{ll}0,& x\text{rational}\hfill \\ x^2,& x\text{irrrational}\hfill \end{array}$

[T] In physics, the magnitude of an electric field generated by a point charge at a distance r in vacuum is governed by Coulomb’s law: $E\left(r\right)=\frac{q}{4\pi {\epsilon }_0{r}^2},$ where E represents the magnitude of the electric field, q is the charge of the particle, r is the distance between the particle and where the strength of the field is measured, and $\frac{1}{4\pi {\epsilon }_0}$ is Coulomb’s constant: $8.988\times {10}^9\text{N}·{\text{m}}^2\text{/}{\text{C}}^2.$

1. Use a graphing calculator to graph $E\left(r\right)$ given that the charge of the particle is $q={10}^{\mathrm{-10}}.$
2. Evaluate $\underset{r\to 0^{+}}{\text{lim}}E\left(r\right).$ What is the physical meaning of this quantity? Is it physically relevant? Why are you evaluating from the right?

[T] The density of an object is given by its mass divided by its volume: $\rho =m\text{/}V.$

1. Use a calculator to plot the volume as a function of density $\left(V=m\text{/}\rho \right),$ assuming you are examining something of mass 8 kg ( $m=8\text{).}$
2. Evaluate $\underset{x\to 0^{+}}{\text{lim}}V\left(r\right)$ and explain the physical meaning.