APPLIED CALCULUS
ONE-SIDED LIMITS AND CONTINUITY
TUTORIAL 1B
ALL COHORTS
10 JANUARY 2024
1.Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is +∞ or −∞
a. lim+(3𝑥 2 − 9)
b. lim− 𝑥(2 − 𝑥)
𝑥→1
𝑥→4
𝑥 2 +4
𝑥+3
e. lim− 𝑥+2
f. lim− 𝑥−2
𝑥→2
i.
lim
𝑥→3+
𝑥→2
√𝑥+1−2
𝑥−3
j. lim+
𝑥→5
l. Find lim − 𝑓(𝑥) and
𝑥→−1
c. lim+ √3𝑥 − 9
𝑥→3
𝑥−√𝑥
g. lim− 𝑥−1
d. lim− √4 − 2𝑥
𝑥→2
h. lim+(𝑥 − √𝑥)
𝑥→1
𝑥→0
√2𝑥−1−3
𝑥−5
lim 𝑓(𝑥) where 𝑓(𝑥) = {
𝑥→−1+
1
, 𝑖𝑓 𝑥 < −1
𝑥−1
2
𝑥 + 2𝑥, 𝑖𝑓 𝑥 ≥ 3
2𝑥 2 − 𝑥, 𝑖𝑓 𝑥 < −1
m. Find lim− 𝑓(𝑥) and lim+ 𝑓(𝑥) where 𝑓(𝑥) = {
𝑥→3
𝑥→3
3 − 𝑥, 𝑖𝑓 𝑥 ≥ −1
2. Decide if the given function is continuous at the specified value of 𝑥.
a. 𝑓(𝑥) = 5𝑥 2 − 6𝑥 + 1 at 𝑥 = 2
b. 𝑓(𝑥) =
𝑥+2
at 𝑥 = 1
𝑥+1
c. 𝑓(𝑥) =
𝑥+1
at 𝑥 = 1
𝑥−1
2𝑥−4
at 𝑥 = 2
3𝑥−2
e. 𝑓(𝑥) =
2𝑥+1
at 𝑥 = 2
3𝑥−6
f. 𝑓(𝑥) =
√𝑥−2
,𝑥 =2
𝑥−4
d. 𝑓(𝑥) =
𝑥 2 −1
, 𝑖𝑓 𝑥 < −1
𝑥+1
at 𝑥 = −1
2
g. 𝑓(𝑥) = 𝑓(𝑥) = {
𝑥 − 3, 𝑖𝑓 𝑥 ≥ −1
3. List all the values of 𝑥 for which the given function is not continuous
a. 𝑓(𝑥) = 3𝑥 2 − 6𝑥 + 9
𝑥 2 −1
e. 𝑓(𝑥) = 𝑥+1
𝑥 2 −2𝑥+1
i. 𝑓(𝑥) = 𝑥 2 −𝑥−2
𝑥+1
b.
𝑓(𝑥) = 𝑥−2
f.
𝑓(𝑥) = (𝑥+3)(𝑥−6)
j.
𝑓(𝑥) = {
3𝑥−2
3𝑥−1
d. 𝑓(𝑥) = 𝑥+1
𝑥
g. 𝑓(𝑥) = (𝑥+5)(𝑥−1)
2𝑥 + 3, 𝑖𝑓 𝑥 ≤ 1
6𝑥 − 1, 𝑖𝑓 𝑥 > 1
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3𝑥+3
c. 𝑓(𝑥) = 2𝑥−6
𝑥
h. 𝑓(𝑥) = 𝑥 2 −𝑥
𝑥 2 , 𝑖𝑓 𝑥 ≤ 2
k. 𝑓(𝑥) = {
9, 𝑖𝑓 𝑥 > 2
3𝑥 − 2, 𝑖𝑓 𝑥 < 0
l. 𝑓(𝑥) = { 2
𝑥 + 𝑥, 𝑖𝑓 𝑥 ≥ 0
2 − 3𝑥, 𝑖𝑓 𝑥 ≤ −1
𝑓(𝑥) = { 2
𝑥 − 𝑥 + 3, 𝑖𝑓 𝑥 > −1
m.
4. A ruptured pipe in a North Sea oil rig produces a circular oil slick that is 𝑦 meters thick at a distance 𝑥
meters from the rupture. Turbulence makes it difficult to directly measure the thickness of the slick at the
source (where (𝑥 = 0), but for 𝑥 > 0, it is found that
𝑦=
0.5(𝑥 2 + 3𝑥)
𝑥 3 + 𝑥 2 + 4𝑥
Assuming the oil slick is continuously distributed, how thick would you expect it to be at the source?
5. In certain situations, it is necessary to weigh the benefits of pursuing a certain goal against the cost of
achieving that goal. For instance, suppose that in order to remove 𝑥% of the pollution from an oil spill, it
costs 𝐶 thousands of dollars, where
𝐶(𝑥) =
12𝑥
100 − 𝑥
a. How much does it cost to remove 25% of the pollution? 50%?
b. Sketch the graph of the cost function
c. What happens as 𝑥 → 100− ? Is it possible to remove all the pollution?
6. A business manager determines that when 𝑥% of her company’s plant capacity is being used, the total
cost of operation is 𝐶 hundred thousand dollars, where
𝐶(𝑥) =
8𝑥 2 − 636𝑥 − 320
𝑥 2 − 68𝑥 − 960
a. Find 𝐶(0) and 𝐶(100)
b. Explain why the result of part (a) cannot be used along with the intermediate value property to show
that the cost of operation is exactly $700,000 when a certain percentage of plant capacity is being used.
7. It is estimated that 𝑡 years from now, the population of a certain suburban community will be 𝑝
7
thousand people. 𝑝(𝑡) = 20 − 𝑡+2
8. An environmental study indicates that the average level of carbon monoxide in the air will be 𝑐 parts
per million when the population is 𝑝 thousand, where
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𝑐(𝑝) = 0.4√𝑝2 + 𝑝 + 21
What happens to the level of pollution 𝑐 in the long run (as 𝑡 → ∞).
9. Find the values of the constant A so that the function 𝑓(𝑥) will be continuous for all 𝑥.
𝐴𝑥 − 3
𝑓(𝑥) = {
3 − 𝑥 + 2𝑥 2
𝑖𝑓 𝑥 < 2
𝑖𝑓 𝑥 ≥ 2
1 − 3𝑥
𝑓(𝑥) = { 2
𝐴𝑥 + 2𝑥 − 3
𝑖𝑓 𝑥 < 4
𝑖𝑓 𝑥 ≥ 4
1
𝑥
10a. Discuss the continuity of the function 𝑓(𝑥) = 𝑥 (1 + ) on the open interval 0 < 𝑥 < 1 and on the
closed interval 0 ≤ 𝑥 ≤ 1
10b. Discuss the continuity of the function
𝑓(𝑥) = {
𝑥 2 − 3𝑥
4 + 2𝑥
𝑖𝑓 𝑥 < 2
𝑖𝑓 𝑥 ≥ 2
on the open interval 0 < 𝑥 < 2 and on the closed interval 0 ≤ 𝑥 ≤ 2
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