12.3 Continuity  (Page 6/10)

$f\left(x\right)=\{\begin{array}{l}5,x\ne 0\\ 3,x=0\end{array}a=0$

$f\left(x\right)=\{\begin{array}{ll}\frac{1}{2-x},\hfill & x\ne 2\hfill \\ 3,\hfill & x=2\hfill \end{array}a=2$

$f\left(x\right)=\{\begin{array}{ll}\frac{1}{x+6},\hfill & x=-6\hfill \\ x^2,\hfill & x\ne -6\hfill \end{array}a=-6$

$f\left(x\right)=\{\begin{array}{ll}3+x,\hfill & x<1\hfill \\ x,\hfill & x=1\hfill \\ x^2,\hfill & x>1\hfill \end{array}a=1$

$f\left(x\right)=\{\begin{array}{ll}3-x,\hfill & x<1\hfill \\ x,\hfill & x=1\hfill \\ 2x^2,\hfill & x>1\hfill \end{array}a=1$

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

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

$f\left(x\right)=\{\begin{array}{ll}\frac{x^2-9}{x+3},\hfill & x<-3\hfill \\ x-9,\hfill & x=-3\hfill \\ \frac{1}{x},\hfill & x>-3\hfill \end{array}a=-3$

$f\left(x\right)=\{\begin{array}{ll}\frac{x^2-9}{x+3},\hfill & x<-3\hfill \\ x-9,\hfill & x=-3\hfill \\ -6,\hfill & x>-3\hfill \end{array}a=3$

$f\left(x\right)=\frac{x^2-4}{x-2},a=2$

$f\left(x\right)=\frac{25-x^2}{x^2-10x+25},a=5$

$f\left(x\right)=\frac{x^3-9x}{x^2+11x+24},a=-3$

$f\left(x\right)=\frac{x^3-27}{x^2-3x},a=3$

$f(x)=\frac{x}{\left|x\right|},a=0$

$f\left(x\right)=\frac{2\left|x+2\right|}{x+2},a=-2$

$f\left(x\right)=x^3-2x-15$

$f\left(x\right)=\frac{x^2-2x-15}{x-5}$

$f\left(x\right)=2\cdot 3^{x+4}$

$f\left(x\right)=\mathrm{-sin}\left(3x\right)$

$f\left(x\right)=\frac{\left|x-2\right|}{x^2-2x}$

$f\left(x\right)=\tan \left(x\right)+2$

$f\left(x\right)=2x+\frac{5}{x}$

$f\left(x\right)={\log }_2\left(x\right)$

$f(x)=\ln x^2$

$f\left(x\right)=e^{2x}$

$f(x)=\sqrt{x-4}$

$f\left(x\right)=\sec \left(x\right)-3$ .

$f\left(x\right)=x^2+\sin \left(x\right)$

Determine the values of $b$ and $c$ such that the following function is continuous on the entire real number line.

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

$x=-3$

$x=2$

$x=4$

Which conditions for continuity fail at the point of discontinuity?

Evaluate $f(0).$

Solve for $x$ if $f(x)=0.$

What is the domain of $f\left(x\right)?$

At what x -coordinates is the function discontinuous?

What condition of continuity is violated at these points?

Consider the function shown in [link] . At what x -coordinates is the function discontinuous? What condition(s) of continuity were violated?

Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at $x=-7$ and $x=1.$

The function $f(x)=\frac{x^3-1}{x-1}$ is graphed in [link] . It appears to be continuous on the interval $\left[-3,3\right],$ but there is an x -value on that interval at which the function is discontinuous. Determine the value of $x$ at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.

Find the limit $\underset{x\to 1}{\lim }f(x)$ and determine if the following function is continuous at $x=1:$

The graph of $f(x)=\frac{\sin (2x)}{x}$ is shown in [link] . Is the function $f\left(x\right)$ continuous at $x=0?$ Why or why not?