Download ABCALC All Limits Homework A (1)

AB Calculus All Limits Homework A
1.
If lim 𝑓(𝑥) = −3, lim 𝑔(𝑥) = 0, and lim ℎ(𝑥) = 8, find the following. If the limit is nonexistent, explain why.
𝑥→𝑎
a)
2.
Name: ________________________________________
𝑥→𝑎
𝑥→𝑎
lim [𝑓(𝑥) + ℎ(𝑥)]
b)
𝑥→𝑎
lim 𝑓(𝑥)2
1
𝑥→𝑎 𝑓(𝑥)
e)
𝑓(𝑥)
𝑥→𝑎 ℎ(𝑥)
g)
𝑓(𝑥)
𝑥→𝑎 𝑔(𝑥)
h)
2𝑓(𝑥)
𝑥→𝑎 ℎ(𝑥) − 𝑓(𝑥)
lim
lim
3
lim √ℎ(𝑥)
𝑥→𝑎
d)
f)
lim
𝑔(𝑥)
𝑥→𝑎 𝑓(𝑥)
lim
lim
50𝜋
)
𝑥
Use the sandwich theorem to show that lim 𝑥 4 𝑐𝑜𝑠 (
𝑥→0
3.
c)
𝑥→𝑎
= 0.
For each of the functions, find lim 𝑓(𝑥), lim 𝑓(𝑥), and identify all horizontal asymptotes.
𝑥→∞
a)
c)
𝑥→−∞
𝑓(𝑥) =
3𝑥 3 − 𝑥 − 1
𝑥+3
b)
𝑓(𝑥) =
3𝑥 + 1
𝑥−4
d)
4.
Evaluate each of the following limits.
a)
2
1
−4
𝑥
+
3
lim
𝑥→5 𝑥 − 5
b)
𝑓(𝑥) =
𝑓(𝑥) =
4𝑥 2 − 3𝑥 + 5
2𝑥 3 + 𝑥 − 1
−2𝑥 2 + 4
√4𝑥 4 + 8𝑥 2 + 1
𝑦 2 + 4𝑦 + 3
𝑦→−3
𝑦2 − 3
lim
c)
e)
4𝑥 + sin 2𝑥
𝑥→0
𝑥
d)
1
𝑥
lim ( −
)
𝑥 𝑥−1
f)
lim
√𝑥 + 6 − 3
𝑥→3
𝑥−3
lim
lim
𝑥→2 (𝑥
𝑥→∞
5.
Given the graph of 𝑓(𝑥) below, find the following limits.
a)
c)
e)
g)
6.
−1
− 2)2
lim 𝑓(𝑥)
b)
𝑥→−4 −
𝑓(−2)
d)
lim 𝑓(𝑥)
f)
𝑓(1)
h)
𝑥→1+
Given the definition of 𝑓(𝑥) below, find the following limits.
5 − 2𝑥,
𝑓(𝑥) = {
4,
4 − 𝑥,
a)
c)
𝑥>1
𝑥=1
𝑥<1
lim 𝑓(𝑥)
b)
lim 𝑓(𝑥)
d)
𝑥→5
𝑥→1+
lim 𝑓(𝑥)
𝑥→1−
lim 𝑓(𝑥)
𝑥→1
lim 𝑓(𝑥)
𝑥→−2
lim 𝑓(𝑥)
𝑥→1−
lim 𝑓(𝑥)
𝑥→1
lim 𝑓(𝑥)
𝑥→−4 +