1. a. removable discontinuity at x=6;
b. jump discontinuity at x=4
2. yes
3. No, the function is not continuous at x=3. There exists a removable discontinuity at x=3.
4. x=6
Solutions to Odd-Numbered Exercises
1. Informally, if a function is continuous at x=c, then there is no break in the graph of the function at f(c), and f(c) is defined.
3. discontinuous at a=−3 ; f(−3) does not exist
5. removable discontinuity at a=−4 ; f(−4) is not defined
7. discontinuous at a=3 ; x→3limf(x)=3, but f(3)=6, which is not equal to the limit.
9. x→2limf(x) does not exist.
11. x→1−limf(x)=4;x→1+limf(x)=1 . Therefore, x→1limf(x) does not exist.
13. x→1−limf(x)=5=x→1+limf(x)=−1 . Thus x→1limf(x) does not exist.
15. x→−3−limf(x)=−6 , x→−3+limf(x)=−31
Therefore, x→−3limf(x) does not exist.
17. f(2) is not defined.
19. f(−3) is not defined.
21. f(0) is not defined.
23. Continuous on (−∞,∞)
25. Continuous on (−∞,∞)
27. Discontinuous at x=0 and x=2
29. Discontinuous at x=0
31. Continuous on (0,∞)
33. Continuous on [4,∞)
35. Continuous on (−∞,∞) .
37. 1, but not 2 or 3
39. 1 and 2, but not 3
41. f(0) is undefined.
43. (−∞,0)∪(0,∞)
45. At x=−1, the limit does not exist. At x=1, f(1) does not exist.
At x=2, there appears to be a vertical asymptote, and the limit does not exist.
47. (x+7)(x−1)x3+6x2−7x
49. fx={x2+42x=1x=1