set 3: limits of functions: a. the intuitive approach (2.2...
TRANSCRIPT
1
Set 3: Limits of functions:
A. The intuitive approach (2.2):
1. Watch the video at:
https://www.khanacademy.org/math/differential-calculus/limit-basics-dc/formal-definition-of-limits-dc/v/limit-
intuition-review
2.
3.
4. Below is the graph of a function )(xf . For each of the given points determine the value of )(af and of
)(lim xfax
. If any of these quantities do not exist, clearly explain why.
2
5. Below is the graph of a function )(xf . For each of the given points determine the value of )(af and of
)(lim xfax
. If any of these quantities do not exist, clearly explain why.
6. Below is the graph of a function )(xf . For each of the given points determine the value of )(af , of
)(lim xfax
, )(lim xfax
and of )(lim xfax
. If any of these quantities do not exist, clearly explain why.
3
7.
.
8. Guess the value of the corresponding limit by using the table method. When possible, verify your result
algebraically (that is, simplify )(xf and calculate the limit of the simpler expression) or graphically (if you have a
graphing device):
a) 1lim2x
b) xx 2lim
c) 2
2lim xx
d) 1
3lim
2
x
x
x e)
2
4lim
2
2
x
x
x f)
x
x
x
39lim
2
0
g) x
e x
x
1lim
5
0
h)
x
x
x
)sin(lim
0 i)
22 2
1lim
xx
j) 2
1lim
2 xx k)
xx
1sinlim
0
4
B. Limit laws (2.3):
1.
(g) )(21)(3lim2
xgxfx
(h) )()(63lim2
xhxfx
2. For each of the following exercises, use the limit properties to calculate the limit. If it is not possible to calculate
the limit, clearly explain why not:
a) 293lim 2
4
xx
x
b) 232
2lim
2
4
2
xx
x
x
c) 63lim 4
2
xx
x
3. Evaluate the limit, if it exists:
a) 3
6lim
2
4
0
x
x
x b)
166
2lim
22
xx
x
x
5
c) d)
e) 1
1lim
3
4
1
x
x
x f)
2
314lim
2
x
x
x
g) x
xx
x
11lim
0 h)
xxxx 20
11lim
i) 21616
4lim
xx
x
x
4.
5. Find the limit, if it exists. If the limit does not exist, explain why:
a) 32lim3
xxx
b) 6
122lim
6
x
x
x c)
x
x
x
2
2lim
2
6
d) 235.0 2
12lim
xx
x
x
e)
xxx
11lim
0
6.
7. Given the function:
2for 145
2for )(
32
xx
xxxxf , evaluate the following limits, if they exist:
a) )(lim3
xfx
and b) )(lim2
xfx
8.
9.
7
C. The Rigorous Definition of Limit (2.4) (Optional):
1. Watch the videos at:
a) https://www.khanacademy.org/math/differential-calculus/limit-basics-dc/formal-definition-of-limits-
dc/v/building-the-idea-of-epsilon-delta-definition
b) https://www.khanacademy.org/math/differential-calculus/limit-basics-dc/formal-definition-of-limits-dc/v/epsilon-
delta-definition-of-limits
and
c) https://www.khanacademy.org/math/differential-calculus/limit-basics-dc/formal-definition-of-limits-
dc/v/proving-a-limit-using-epsilon-delta-definition
2. Explain as precisely (as rigorously) as you can what each of the following means and illustrate with a sketch:
3. Describe several ways in which a limit can fail to exist. Illustrate with sketches.
4. Determine whether the following statement is true or false. If it is true, explain why. If it is false, explain why or
give an example which disproves the statement:
Let )(xf be a function such that 6)(lim0
xfx
. Then it is true that there exists a number 0 such that if
x0 then 16)( xf .
8
5.
What does this graph and your calculations suggest about )(lim1
xfx
?
6.
What does this graph and your calculations suggest about )(lim3
xfx
?
9
7.
What does this graph and your calculations suggest about 2
1lim xx
?
8. Prove the following statements using the rigorous - definition of a limit:
a) 23
42lim
1
x
x b) 5
5
43lim
10
x
x
c) axax
lim d) 0lim 2
0
x
x
e) 172lim 2
2
xx
x
10
D. Continuity (2.5):
1. Write an equation that expresses the fact that a function )(xf is continuous at the number 4.
2. If )(xf is continuous on , , what can you say about its graph?
3.
4. Sketch the graph of a function )(xf that is continuous except at the stated discontinuity:
a) discontinuous at -1 and 4, but continuous from the left at -1 and from the right at 4 ;
b) has a removable discontinuity at 3 and a non-removable (jump) discontinuity at 5 .
5. The graph of a function )(xf is shown below. Based on this graph, determine where the function is
discontinuous:
a)
11
b)
6. Use the definition of continuity and the methods to calculate limits to determine if the given function is
continuous or discontinuous at the given points. Support your work:
a)
12
b)
c)
d)
e)
7. Determine where the given function is discontinuous:
a) 7132
211)(
2
xx
xxf b)
xxx
xxf
23
2
6
1)(
13
c)
12
cos
14)(
x
xxf d)
2 if 2
20 if 2
0 if 1
)(
2
2
xx
xx
xx
xf
8. Find Ra such that the function
3 if
3 if 3)(
xxa
xxxf is continuous for 3x .
9.
10.
11. Use the Intermediate Value theorem to show that there is at least one root of the specified equation in the given
interval. Support your work by showing the continuity of the appropriate function:
a) 8,4on 071 43 xx
b) 15,-5-on 3112 xx
c) 5,1-on 08
152
x
xx
d) 1,2-on 04ln12ln 22 xx
e) 4,0on 510 23 xexx
14
f) 0,1on 23 xex
g) 1,2on x -)sin( 2xx
E. Limits at infinity and infinite limits. Horizontal Asymptotes and Vertical asymptotes. (2.6 and 2.2).
1. Explain as precisely as you can the meaning of:
a)
)(lim3
xfx
b)
)(lim4
xfx
c) 5)(lim
xfx
d) 3)(lim
xfx
2.
15
3.
4.
5. Calculate the following infinite limits. Support your work:
a) 3
2lim
3
x
x
x b)
3
2lim
3
x
x
x
c) 3
2lim
3
x
x
x d)
65
82lim
2
2
2
xx
xx
x
e) 65
82lim
2
2
2
xx
xx
x f)
8
1lim
32
x
x
x
16
6. Find the equations of all the vertical asymptotes for the given function:
a) x
xf
9
6)( b)
2
2
23
1)(
xx
xxf
c) x
xxf
)4sin()( d)
x
xxf
)4tan()(
7. Calculate the following limits. Support your work:
a) 12
23lim
x
x
x b)
1
2lim
2
x
x
x
c)
xxx
x
x
22
22
1
12lim d) xx
1lim
e) 1
1lim
2
1
x
x
x f) 2
22lim
2
x
x
x
g) x
xx
x 2cos
)sin()cos(lim
4
h) 1
132lim
3
4
xx
xx
x
i) xxxx
2lim 2
j) 1lim 2
xx
k) 1lim 2
xx
l)
2
331
limx
xx
x
m) xxx
2lim n) 1
)(sinlim
2
2
x
x
x
8. Find the horizontal asymptotes of each curve:
a) 2
12)(
x
xxf b)
42
41)(
xx
xxf
c) 53
12)(
2
x
xxf
17
F. Derivatives and Rates of Change (2.7):
1. Watch the video at:
https://www.khanacademy.org/math/differential-calculus/derivative-intro-dc/intro-to-diff-calculus-dc/v/newton-
leibniz-and-usain-bolt
2. Solve 10 exercises from:
https://www.khanacademy.org/math/differential-calculus/derivative-intro-dc/intro-to-diff-calculus-dc/e/graphs-of-
functions-and-their-derivatives
3. Write an expression for the slope of the tangent line to the curve )(xfy at the point )(, afa .
4. Suppose that an object moves along a straight line with position )(tf at time t . Write an expression for the
instantaneous velocity of the object at time at . How can you interpret this velocity in terms of the graph of )(tf ?
5. Find the slope of the tangent line to the parabola 24 xxy at the point (1,3 ). Find an equation of this tangent
line.
6. Find the slope of the tangent line to the curve 3xxy at the point (1,0 ). Find an equation of this tangent line.
7. Find an equation of the tangent line to the curve 2
12
x
xy at the point (1,1) .
8. a) Find the slope of the tangent line to the curve 32 243 xxy at the point where ax .
b) Using part a), find equations of the tangent lines at the points (1,5) and (2,3).
c) Using a graphical device, graph the curve and both tangents on a common screen.
18
9.
10.
a) describe and compare how the runner run the race in terms of their instantaneous speeds.
b) at what time is the distance between the runners the greatest?
c) at what time do they have the same instantaneous speeds?
19
11. If a ball is thrown into the air with an initial velocity of sec/ 40 ft , its height (in feet) after t seconds is given by:
21640)( ttty . Find its velocity when 2t .
12. Find )(' af for:
a) 143)( 2 xxxf b) xxxf 32)(
c) 2)( xxf
G. The derivative as a function (2.8):
1. Write a formula for the derivative )(' xf of a function )(xf . Can you write another formula for )(' xf ?
2.
20
3. The graph of a function )(xf is given. For each function, sketch the graph of its derivative )(' xf directly
underneath it:
a) b)
c) d)
4. Calculate )(' xf using your definition from exercise 1. State the domain of )(xf and the domain of )(' xf in each
case:
a) 3
1
2
1)( xxf b) xxf )(
21
c) x
xf1
)( d) x
xy
3
21
5.
a) What does it mean for )(xf to be differentiable at a ?
b) What is the relationship between differentiability and continuity for a function?
c) Sketch the graph of a function which is continuous but not differentiable at 2a .
d) Describe several ways in which a function can fail to be differentiable. Illustrate each case with skecthes.
6. The graph of )(xf is given. In each case, state with reasons all points at which )(xf is not differentiable:
a) b)
c) d)
7.
22