Math 121 Sample Test 2 – Oct 2016
Question 1 Find the following limits
(a)
2x2 − 4
x→−∞ x − 6
lim
(c)
x2 − 4
lim
x→∞ 4x2 + x − 6
(d)
sin x
x→0 x2
lim
1
(e)
lim +
x→−1
(f )
|x − 1|
x2 − 1
cos x
.
x→∞ x2
lim
2
Question 2:
Find (if any) a value of a > 0 such that f (x) is continuous for all x.
(
2x
if x < 2
f (x) =
3 − sin(a) if x ≥ 2.
Question 3 (Technique. Basic Rules only!):
Differentiate
(1) f (t) = (t − 1)2
(2) f (x) =
3
x−1
x+1
(3) f (x) =
A
x−a
(4) f (t) = sec x csc x
(5) f (x) =
(6) f (x) =
4
x−1
tan x
2x cos x
x−1
(7) f (x) = x(2x − 1) tan x
π
(8) f (x) = √
3
x x5
5
Question 4(High order derivatives):
Let f (x) =
a+x
,
a−x
00
where a is a constant. Compute f (0).
Question 5 (Asymptotes):
Find all asymptotes of
f (x) =
and make a sketch of f(x).
6
x
2x + 1
Question 6(Differentiability):
Let f (x) = |x2 − 2x|. Find all points where f (x) fails to be differentiable.
Question 7(Definition):
Let f (x) =
3
.
x−1
0
Use the definition of the derivative to compute f (0).
7
Question 8(Tangent):
Find all points on the graph of f (x) = x3 − x2 at which the tangent line is horizontal.
8