C Roettger, Spring 15
Name (please print): . . . . . . .
Math 165 - Practice Exam 1
State clearly what your result is. Show your work. No calculators, notes
or books. Time is 50 minutes. Please write your name on the back and front
of this paper. Give exact results.
Problem 1 Evaluate the following limits.
a)
b)
c)
lim cos(π(x + 1))
x→0
tan(4y 2 )
y→0
3y 2
cos(2πt) − 1
lim
t→1
t−1
lim
In c), you may use the addition formula of the cosine,
cos(a + b) = cos a cos b − sin a sin b
Problem 2 Evaluate the limits, if they exist.
a)
b)
c)
d)
e)
5t4 − 2t3 + 9
t→∞
2t4 + 100
3 − 2x + 4x3
lim
x→∞
3x2 + 2x
5 − 2u
lim 2
u→3 u − 9
5 − 2v
lim+ 2
v→3 v − 9
2w4 − sin(w)
lim
.
w→∞
w5 + 1
lim
Problem 3 Suppose the functions f (x) and g(x) are continuous everywhere and have these values: f (0) = 1, f (1) = 5, f (2) = 4, g(0) = 2,
g(2) = 1, g(4) = 3. Find limx→2 f (x) + g(x) and limx→2 f (g(x)).
Problem 4 a) Find the instantaneous rate of change of
f (x) = 2x2 + 5x
at x = −1. Use a limit computation, no rules for derivatives that are not
covered in Chapter 2.
b) Find an equation for the tangent line to the graph of f (x). Make a sketch
of the graph and this tangent line, using the values
x −3 −2 −1 0 1
f (x)
3 −2 −3 0 7
Problem 5 Evaluate the following limit.
p
sin(4t2 )
lim+
t→0
t
Problem 6 Find the limit
√
√
x2 + x + 1 − x2 − 3x + 1
L = lim
.
x→0
x
Problem 7 Consider the function f (x) defined below. What value can
be assigned to the constant a to make f (x) continuous everywhere?
2 + sinx x for x < 0,
f (x) =
x2 + a for x ≥ 0.