2.2 The limit of a function  (Page 8/12)

$\underset{t\to 0^{+}}{\text{lim}}\frac{\text{cos}t}{t}$

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

$\underset{\theta \to 0}{\text{lim}}\text{sin}\left(\frac{\pi }{\theta }\right)$

$\underset{\alpha \to 0^{+}}{\text{lim}}\frac{1}{\alpha }\text{cos}\left(\frac{\pi }{\alpha }\right)$

$\underset{x\to 10}{\text{lim}}f\left(x\right)=0$

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

$\underset{x\to \mathrm{-8}}{\text{lim}}f\left(x\right)=f\left(\mathrm{-8}\right)$

$\underset{x\to 6}{\text{lim}}f\left(x\right)=5$

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

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

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

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

$f\left(1\right)$

$\underset{x\to 0^{-}}{\text{lim}}f\left(x\right)$

$\underset{x\to 0^{+}}{\text{lim}}f\left(x\right)$

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

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

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

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

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

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

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

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

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

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

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

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

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

$\underset{x\to 0}{\text{lim}}h\left(x\right)$

$\underset{x\to 0^{-}}{\text{lim}}f\left(x\right)$

$\underset{x\to 0^{+}}{\text{lim}}f\left(x\right)$

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

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

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

$\underset{x\to 2}{\text{lim}}f\left(x\right)=1,\underset{x\to 4^{-}}{\text{lim}}f\left(x\right)=3,\underset{x\to 4^{+}}{\text{lim}}f\left(x\right)=6,x=4$ is not defined.

$\underset{x\to -\infty }{\text{lim}}f\left(x\right)=0,\underset{x\to {\mathrm{-1}}^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty ,$ $\underset{x\to {\mathrm{-1}}^{+}}{\text{lim}}f\left(x\right)=\infty ,\underset{x\to 0}{\text{lim}}f\left(x\right)=f\left(0\right),f\left(0\right)=1,\underset{x\to \infty }{\text{lim}}f\left(x\right)=\text{−}\infty$

$\underset{x\to -\infty }{\text{lim}}f\left(x\right)=2,\underset{x\to 3^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty ,$ $\underset{x\to 3^{+}}{\text{lim}}f\left(x\right)=\infty ,\underset{x\to \infty }{\text{lim}}f\left(x\right)=2,f\left(0\right)=\frac{\mathrm{-1}}{3}$

$\underset{x\to -\infty }{\text{lim}}f\left(x\right)=2,\underset{x\to \mathrm{-2}}{\text{lim}}f\left(x\right)=\text{−}\infty ,$ $\underset{x\to \infty }{\text{lim}}f\left(x\right)=2,f\left(0\right)=0$

$\underset{x\to -\infty }{\text{lim}}f\left(x\right)=0,\underset{x\to {\mathrm{-1}}^{-}}{\text{lim}}f\left(x\right)=\infty ,\underset{x\to {\mathrm{-1}}^{+}}{\text{lim}}f\left(x\right)=\text{−}\infty ,$ $f\left(0\right)=\mathrm{-1},\underset{x\to 1^{-}}{\text{lim}}f\left(x\right)=\text{−}\infty ,\underset{x\to 1^{+}}{\text{lim}}f\left(x\right)=\infty ,\underset{x\to \infty }{\text{lim}}f\left(x\right)=0$

Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, x , is shown here. We are mainly interested in the location of the front of the shock, labeled $x_{\text{SF}}$ in the diagram.

1. Evaluate $\underset{x\to x_{SF}{}^{+}}{\text{lim}}\rho \left(x\right).$
2. Evaluate $\underset{x\to x_{SF}{}^{-}}{\text{lim}}\rho \left(x\right).$
3. Evaluate $\underset{x\to x_{SF}}{\text{lim}}\rho \left(x\right).$ Explain the physical meanings behind your answers.

A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where x is the position in meters of the runner and t is time in seconds. What is $\underset{t\to 2}{\text{lim}}x\left(t\right)?$ What does it mean physically?