Solutions for Continuity
Solutions to Try Its
1. a. removable discontinuity at [latex]x=6[/latex]; b. jump discontinuity at [latex]x=4[/latex] 2. yes 3. No, the function is not continuous at [latex]x=3[/latex]. There exists a removable discontinuity at [latex]x=3[/latex]. 4. [latex]x=6[/latex]Solutions to Odd-Numbered Exercises
1. Informally, if a function is continuous at [latex]x=c[/latex], then there is no break in the graph of the function at [latex]f\left(c\right)[/latex], and [latex]f\left(c\right)[/latex] is defined. 3. discontinuous at [latex]a=-3[/latex] ; [latex]f\left(-3\right)[/latex] does not exist 5. removable discontinuity at [latex]a=-4[/latex] ; [latex]f\left(-4\right)[/latex] is not defined 7. discontinuous at [latex]a=3[/latex] ; [latex]\underset{x\to 3}{\mathrm{lim}}f\left(x\right)=3[/latex], but [latex]f\left(3\right)=6[/latex], which is not equal to the limit. 9. [latex]\underset{x\to 2}{\mathrm{lim}}f\left(x\right)[/latex] does not exist. 11. [latex]\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)=4;\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)=1[/latex] . Therefore, [latex]\underset{x\to 1}{\mathrm{lim}}f\left(x\right)[/latex] does not exist. 13. [latex]\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)=5\ne \underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)=-1[/latex] . Thus [latex]\underset{x\to 1}{\mathrm{lim}}f\left(x\right)[/latex] does not exist. 15. [latex]\underset{x\to -{3}^{-}}{\mathrm{lim}}f\left(x\right)=-6[/latex] , [latex]\underset{x\to -{3}^{+}}{\mathrm{lim}}f\left(x\right)=-\frac{1}{3}[/latex] Therefore, [latex]\underset{x\to -3}{\mathrm{lim}}f\left(x\right)[/latex] does not exist. 17. [latex]f\left(2\right)[/latex] is not defined. 19. [latex]f\left(-3\right)[/latex] is not defined. 21. [latex]f\left(0\right)[/latex] is not defined. 23. Continuous on [latex]\left(-\infty ,\infty \right)[/latex] 25. Continuous on [latex]\left(-\infty ,\infty \right)[/latex] 27. Discontinuous at [latex]x=0[/latex] and [latex]x=2[/latex] 29. Discontinuous at [latex]x=0[/latex] 31. Continuous on [latex]\left(0,\infty \right)[/latex] 33. Continuous on [latex]\left[4,\infty \right)[/latex] 35. Continuous on [latex]\left(-\infty ,\infty \right)[/latex] . 37. 1, but not 2 or 3 39. 1 and 2, but not 3 41. [latex]f\left(0\right)[/latex] is undefined. 43. [latex]\left(-\infty ,0\right)\cup \left(0,\infty \right)[/latex] 45. At [latex]x=-1[/latex], the limit does not exist. At [latex]x=1[/latex], [latex]f\left(1\right)[/latex] does not exist. At [latex]x=2[/latex], there appears to be a vertical asymptote, and the limit does not exist. 47. [latex]\frac{{x}^{3}+6{x}^{2}-7x}{\left(x+7\right)\left(x - 1\right)}[/latex] 49. [latex]fx=\begin{cases}x^{2}+4 \hfill& x\neq 1 \\ 2 \hfill& x=1\end{cases}[/latex]Licenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution.
