Exercise Set 1
1. Find an equation of the line passing through the points (2, 1) and (4, −1)
(Answer: x + y = 3)
2. Find an equation of the line passing through the point (1, 2) whos slope is 3
(Answer: y = 3x − 1)
3. Find an equation of the line passing through (2, 3) that is perpendicular to the line 2x − 3y = 4.
(Answer: 3x + 2y = 12)
In exercises 4-12, find the largest possible domain of the given function.
√
4. f (x) = x2 − 2x − 8
5. g(t) = ln(t3 − 2t + 1)
√
6. h(s) = 3 s5 − 3s2 + 2s
x2 +1
7. k(x) = e x2 +4x+3
8. m(y) =
y 4 +2y 3 −5y 2 +2y−3
y 3 +2y 2 −5y−6
9. n(s) = sin(s2 − 4)
10. p(t) = tan(2t + π)
11. r(x) = 3x155 − 5x23 + 25x2 − 2017x + 3
12. f (z) = sec( π2 z)
In exercises 13-17, sketch the graph of the given parabola (a) by shifting, reflecting and scaling
the graph of y = x2 , (b) by finding vertex and intercepts.
13. y = x2 − 2x − 2
14. y = x2 + 2x − 3
15. y = 12 − x2 − 4x
16. y = 2x2 − 8x + 6
17. y = x2 + 4x + 3
In exercises 18-20, sketch the graph of the given function by shifting, reflecting and scaling
y = |x|
18. f (x) = 4 − |x + 2|
19. g(x) = |2x − 4| + 2
20. h(x) = 3 |x − 2| − 4
In exercises 21-45, evaluate the limits
21. lim x3 − 2x2 + 4x − 5
x→2
x4 − 2x2 + 1
x→−1 x3 − 5x2 − 4x + 2
22. lim
x2 − 9
x→3 x3 − 27
23. lim
x2 + 2x − 3
x→−3 x2 + 5x + 6
24. lim
|x2 − 2x| + 4x
x→0 |2x + 3| − |x2 − 3x − 3|
25. lim
sin(x2 − 1)
x→1
|x − 1|
26. lim
1 − cos(x + 2) + tan(x2 − 4)
x→−2
sin(2x − 4)
√
x−2
28. lim 2
x→4 x + x − 20
√
3
x+3
29. lim
x→−27 sin(x + 27)
√
x−8
30. lim √
x→64 3 x − 4
27. lim
x2 − 1
x→1 x3 − x2 − x + 1
31. lim
32. lim+
x3 − 2x2 + x − 4
x3 − 2x − 4
33. lim−
x3 − 2x2 + x − 4
x3 − 2x − 4
x→2
x→2
2x + 3x2 − 4
x→−∞ 4x2 − 2x + x3
34. lim
x2 − 3x + 4x4
x→∞ 2x3 + 3x4 − 5x − 4
35. lim
x4 − 2x + 5
36. lim 3
x→−∞ x + 2x2 + 1
√
37. lim
x2 + x − 5 − x + 1
x→∞
38. lim
x→−∞
39. lim
√
x→∞
40. lim
3x + 5 +
x→−∞
2x2
√
√
9x2 + 21x − 8
√
2
− 5x − x + 3
4x2 + x + 3 − 2x + 1
1
41. lim x cos
x→0
x
sin2 x
x→∞ ln x
42. lim
1 − cos x
x→−∞
e−x
43. lim
sin(3x) tan(2x) + 5x2
x→0
x2 sin(5x) cot(7x)
2
1
45. lim x − 1 cos
x→1
x3 − 1
44. lim
46. Let

2x + 4




0



2

x +2



3
f (x) =
2x + 1




0




3x
−2



3x + 2
if
x < −2
if
x = −2
if −2 < x < 1
if
x=1
if 1 < x < 3
if
x=3
if 3 < x < 4
if
4<x
Find all discontinuities of f (x) and classify them as removable or jump discontinuity.
In exercises 47-50, find all vertical, horizontal and slant asymptote(s) of the given function, if
exists.
47. f (x) =
x3 −4x
x2 +4x−12
48. g(t) = e1/t
49. h(s) = ln(x2 − 6x + 9) − ln(x2 + 3)
50. p(x) =
3x2 +2x+5
x2 −5x+6