Continuity — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}2x+3,& x\ne 1\\ k,& x=1\end{array}$$

   Answer: $5$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to 1}{\lim }2x+3=5$$
2. 2.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}{x}^2-1,& x\ne 2\\ k,& x=2\end{array}$$

   Answer: $3$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to 2}{\lim }{x}^2-1=3$$
3. 3.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}3x+5,& x\ne -1\\ k,& x=-1\end{array}$$

   Answer: $2$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to -1}{\lim }3x+5=2$$
4. 4.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}{x}^2-2x,& x\ne 3\\ k,& x=3\end{array}$$

   Answer: $3$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to 3}{\lim }{x}^2-2x=3$$
5. 5.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}{x}^3-x,& x\ne 2\\ k,& x=2\end{array}$$

   Answer: $6$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to 2}{\lim }{x}^3-x=6$$
6. 6.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}{x}^2+4x+1,& x\ne -2\\ k,& x=-2\end{array}$$

   Answer: $-3$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to -2}{\lim }{x}^2+4x+1=-3$$
7. 7.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}5x+2,& x\ne 0\\ k,& x=0\end{array}$$

   Answer: $2$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to 0}{\lim }5x+2=2$$
8. 8.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}{x}^2+3,& x\ne 1\\ k,& x=1\end{array}$$

   Answer: $4$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to 1}{\lim }{x}^2+3=4$$
9. 9.

   Choose k so the function is continuous.

   $$f(x)=\{\begin{array}{ll}2x-7,& x\ne 4\\ k,& x=4\end{array}$$

   Answer: $1$

   Step-by-step solution

   1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
   2. $$k=\underset{x\to 4}{\lim }2x-7=1$$
10. 10.

    Choose k so the function is continuous.

    $$f(x)=\{\begin{array}{ll}{x}^2+x,& x\ne -3\\ k,& x=-3\end{array}$$

    Answer: $6$

    Step-by-step solution

    1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
    2. $$k=\underset{x\to -3}{\lim }{x}^2+x=6$$
11. 11.

    Choose k so the function is continuous.

    $$f(x)=\{\begin{array}{ll}{x}^3+2x,& x\ne 2\\ k,& x=2\end{array}$$

    Answer: $12$

    Step-by-step solution

    1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
    2. $$k=\underset{x\to 2}{\lim }{x}^3+2x=12$$
12. 12.

    Choose k so the function is continuous.

    $$f(x)=\{\begin{array}{ll}{x}^2-4x,& x\ne 5\\ k,& x=5\end{array}$$

    Answer: $5$

    Step-by-step solution

    1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
    2. $$k=\underset{x\to 5}{\lim }{x}^2-4x=5$$
13. 13.

    Choose k so the function is continuous.

    $$f(x)=\{\begin{array}{ll}4x^2+3,& x\ne -1\\ k,& x=-1\end{array}$$

    Answer: $7$

    Step-by-step solution

    1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
    2. $$k=\underset{x\to -1}{\lim }4x^2+3=7$$
14. 14.

    Choose k so the function is continuous.

    $$f(x)=\{\begin{array}{ll}{x}^3-5,& x\ne 3\\ k,& x=3\end{array}$$

    Answer: $22$

    Step-by-step solution

    1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
    2. $$k=\underset{x\to 3}{\lim }{x}^3-5=22$$
15. 15.

    Choose k so the function is continuous.

    $$f(x)=\{\begin{array}{ll}2x^2+x-3,& x\ne 2\\ k,& x=2\end{array}$$

    Answer: $7$

    Step-by-step solution

    1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
    2. $$k=\underset{x\to 2}{\lim }2x^2+x-3=7$$
16. 16.

    Choose k so the function is continuous.

    $$f(x)=\{\begin{array}{ll}{x}^3+3x^2,& x\ne -2\\ k,& x=-2\end{array}$$

    Answer: $4$

    Step-by-step solution

    1. $$\text{Continuity requires }f(c)=\underset{x\to c}{\lim }f(x).$$
    2. $$k=\underset{x\to -2}{\lim }{x}^3+3x^2=4$$

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