Description
Problem 1
(10 points)
a) Compute the following limit lim
φ→0
1 − cos2 φ
φ2
. (5 points)
b) Find the asymptotes of the graph 3x
2−12x+9
x2−5
. (5 points)
Problem 2
(10 points)
Find the limit of the following functions using ”Squeeze Law”.
a) limx→0 x
n
cos( 1
xn ) where n ∈ N \ {0} (5 points)
b) limx→0 x
2
e
sin( 1
x
)
(5 points)
Problem 3
(10 points)
Show that the equation
cos(x) = e
x + x + 2
has at least one solution over R. (10 points)
Hint : Intermediate value theorem.
Bonus: Prove that the function
f(x) = e
x + x + 2
has only one root (5 points)