Math 165 - Assignment #1
Due 6/24/2013
This assignment covers sections 1,3,4,5,6 from chapter 1.
1.- Find the indicated limit.
(a) lim
x→2
x3 −16x
x−4
√
(b) lim
x→7+
(x−11)3
x−11
(x+h)2 −x2
h
h→0
(c) lim
2.- For the function f graphed below, find the indicated limit or function value, or state that it does not exist.
y
3
2
1
x
−3
(a) f (−2)
(b)
lim f (x)
x→−2−
(c) lim f (x)
x→−2
(d)
lim f (x)
x→−2+
(e) f (2)
−2
−1
1
2
3
(f) lim f (x)
x→2
3.- Sketch the graph of f (x) = JxK − x; then find each of the following or state that it does not exist.
(a) f (1)
(b) lim f (x)
x→1
(c) lim f (x)
x→1−
(d)
lim f (x)
x→1/2
4.- Use Theorem A to find each of the limits. Justify each step by appealing to a numbered statement, as in Examples
1-4 in the textbook.
√
(a) lim −x2 − 4x3
x→−2
(b) lim
x→0−
4x2 +444x−44
x2 −4x−4
5.- Find the limit if lim f (x) = 5 and lim g(x) = −2.
x→a
x→a
lim 2f (x)−5g(x)
x→a f (x)+2g(x)
6.- Find each of the right-hand and left-hand limits or state that they do not exist.
√
(a)
lim +
x→−3
3+x
x
(b) lim
x→3+
√x−3
x2 −9
7.- Evaluate each limit.
cos(x)
x→π x+1
(a) lim
(b) lim
sin(2x)
3x
(c) lim
sin(4x)−3x
x sec(x)
x→0
x→0
8.- Plot the functions u(x) = |x|, l(x) = −|x| and f (x) = x sin(1/x3 ). Then use these graphs along with the squeeze
Theorem to determine the limit of f (x) as x → 0.
9.- Find the limits.
cos2 (x)
2
x→∞ x −7
(a) lim
(b)
3x3 +x
2 +7x−4
2x
x→−∞
lim
(c) lim
x→3−
x
x−3
10.- Find the horizontal and vertical asymptotes for the graphs of the function f (x) =
2
x−3 .
Then sketch their graphs.
11.- Is the function below continuous at x = 3? Justify your answer using the limit concept.
f (x) =
 3
x −8

 x−2
if x 6= 2


if x = 2.
13
12.- The given function is not defined at a certain point. How should it be defined in order to make it continuous at
that point?
√
f (x) =
13.- At what points, if any, are the functions discontinuous?
(a) f (x) =
22−x2
x2 −5x−πx+5π
(b) g(x) =
√ 1
9+x2
√
x− 2
x−2