MATH1210: Midterm 1 Practice Exam
The following are practice problems for the first exam.
1. Sketch a function f satisfying the following conditions
• The domain of f is [0, 4]
• f (0) = f (1) = f (2) = f (3) = f (4) = 1
• lim f (x) = 2
x→1
• lim f (x) = 1
x→2
• lim− f (x) = 2
x→3
• lim f (x) = 1
x→3+
2. Estimate lim f (x) from the following table of functional values
x→5
x
4.75
4.9
4.99
4.999
5
5.001
5.01
5.1
5.25
f(x)
-.5
-.2
-.02
-.002
17
.002
.02
.2
.5
3. Compute the following limits
p
(a) lim
5x2 + 2x
x→−3
(w − 2)(w2 − w − 6)
w→2
w2 − 4w + 4
x2 + x
(c) lim 2
x→−1 x + 1
(b) lim
4. Assume that lim f (x) = 1 and that lim g(x) = π. Compute
x→a
x→a
p
(a) lim [ g(x) + 2f (x)]
x→a
g 2 (x) − f (x)
x→a g(x) + f (x)
(b) lim
3
(c) lim [f (x) + g(x)]
x→a
5. Show that lim x2 cos (1/x) = 0. (Hint: Use the squeeze theorem.)
x→0
1
6. Compute the following limits
tan 2t
sin 2t − 1
sin 6x
(b) lim
x→0 tan 9x
sin2 2t
(c) lim
θ→0
3t
cos 2t
(d) lim
5t
t→0+
(a) lim
t→0
7. Compute the following limits AT INFINITY AND BEYOND
3x2
x→∞ x2 − 8x + 100
πθ5
(b) lim 5
θ→−∞ θ − 5θ 4
n
(c) lim
n→−∞ n2 + 1
(a) lim
8. Compute the following (possibly infinite) limits
t2
t→π + sin t
x2 + 2x − 8
(b) lim
x2 − 4
x→2+
(a) lim
9. Let f (x) =
(x − 3)(x2 + 2x + 2)
. Compute the following
x2 + 2x − 15
(a) lim f (x)
x→3
(b) lim+ f (x)
x→5
(c) lim f (x)
x→5−
10. Compute the following limits
2x
|x|
sin x
(Hint: The squeeze theorem works for limits at infinity.)
(b) lim
x→∞ x
(c) lim x sin(1/x) (Hint: lim f (x) = lim f (1/x))
(a) lim−
x→0
x→∞
x→∞
x→0
11. Let f (x) be determined by the graph shown here:
2
At each of the following points, determine if f is continuous. If so, is the discontinuity
removable or non-removable?
(a) x = −3
(b) x = −2
(c) x = −1
(d) x = −2
12. At what points (if any) is the function


x
f (x) = x2


2−x
if x < 0
if 0 ≤ x ≤ 1
if x > 1
discontinuous?
13. Show that x3 + 3x = 2 has a solution between 0 and 1. (Hint: Intermediate value theorem)
3