Math 125 Review for Midterm 1 Spring 2016
1. The graph of the function f is pictured below. Find the following limits or explain why they
do not exist.
(a) lim f (x) = . (b) lim f (x) = . (c) lim f (x) = . (d) lim f (x) = . (e) lim f (x) = .
x→3−
x→3+
x→3
x→−3+
x→−3−
(f) lim f (x) = . (g) List all numbers in the open interval (−6, 6) at which f is not continuous.
x→−3
6
6
4
c
2
s
0
s
-2
c
-4
-6
-6
-4
-2
0
2
4
6
2. Complete the following definition of a limit:
Let f (x) be defined on an open interval about c, except possibly at c itself. We say that the limit
of f (x) at x approaches c is the number L, and write lim f (x) = L, if, for every > 0, there
x→c
exists a number δ > 0 such that for all x,
3.
Solve sin(2θ) − cos(θ) = 0 for the angle θ, where 0 ≤ θ < 2π. Show all your work.
There might be multiple solutions.
4.
Prove the limit statement lim 4x − 7 = 5. That is, for all > 0, find δ > 0 (depending
x→3
on epsilon), such that, for all x satisfying 0 < |x − c| < δ the inequality |f (x) − L| < holds.
4. Find each of the following limits. You must show all of your work to receive full credit.
√
√
|x − 1|
x2 + 3x − 10
1− h+1
2 − x2 − 5
1 + θ + sin θ
(a) lim
(b) lim
(c) lim
(d) lim
(e) lim
+
x→−5
x→−3
h→0
θ→0
x
−
1
x
+
5
h
x
+
3
3 cos θ + 1
x→1
p
5. Given the function 2x2 − 18, find all c at which f (x) is (a) continuous, (b) right-continuous
but not continuous (c) left-continuous but not continuous.
6. Find each of the following limits. You must show all of your work to receive full credit.
√
√
√
7x
x2 + 3x3 − 16
x4 − 16
(a) lim
(b) lim
(c)
lim
x
−
16
−
x
−
9
(d)
lim
x→∞
x→∞ 5x2 + 16x − 4
x→−∞ 5x3 + 9x2 − x + 16
x→0 tan 5x
7.
Use the Intermediate Value Theorem to show that the equation x3 − 6x2 + 8x + 1 = 0
has three real solutions