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]