1
Chapter 2 Limits
2.1
1. Simplify each expression:
(a)
k 4  x4
x2  k 2
(b)
x
1  2x

x  3x  18 x  3
2
2. Graph the following piece-wise functions.
 x  2,
(a) f ( x)  
2  x,
2
x0
x0

cos x,
(b) f ( x)  
sin x,



2
x0
0 x

2
3. Given the following function sketch the tangent lines to the graph when x  3,  1, 0, 2, 3.4 and estimate the slope of the tangent line.
2
4. The graph sketched below is the function
y  f ( x)  x . Draw the secant line between the points whose x – coordinates
3
are x  0 and x  2 , and find the slope of the secant line. Then continue to draw secant lines through the following points whose x –
coordinates are as follows and compute the slope of each line.
i) x  .5 and x  2
ii) x  1
and x  2
iii) x  1.5 and x  2
iv) x  1.7 and x  2
v) x  1.9 and x  2
Based on the slopes of the secant lines, can you guess what the slope of the tangent line at
x  2 of the curve will be?
f (b)  f (a )
 average velocity , which is the slope of the secant line, where f is continuous on [a, b]. By definition the
ba
position
average velocity 
=change in position over the change in time.
time
Ave. rate 
5. The path of a particle is given by the curve
Estimate the instantaneous velocity as
f  t   t where t  0 . Find the average rate (velocity) of the particle on the given intervals.
t  2 hours. The units of the y –axis consists of miles.
(a) Average vel. [ 0, 2] =
(b) Average vel. [0.5, 2] =
(c) Average vel. [1, 2] =
(d) Average vel. [1.5, 2] =
(e) Average vel. [1.8, 2] =
(f) Instantaneous velocity as
2.2 Limits of Functions
1. using a table, evaluate each limit.
(a)
lim ( x 2  2 x) 
x 2
x
f(x)
1.923
1.95
1.999
2
2.001
2.10
2.15
t 2 =
3
6
2. Find the following limits. Use the graph.
4
f
2
-5
5
-2
10
The following points are not
on the graph. (2, 0), (4, 2),
(4, 4). The point (4, 3) is on
the graph.
-4
(a)
lim f  x  
(d)
lim f  x  
-6
(b)
x 0
(e)
x 3
lim f  x  
(c)
lim f  x  
(f)
x  2
x  4
lim f  x  
x  2
lim f  x  
x  4
3. Find each limit. Note: the points (0, ½) and (2, 2) are not on the graph of the function f . Any vertical asymptotes?
(a)
(d)
lim f  x  =
(b)
lim f  x  =
(e)
x 0
x 2
lim f  x  =
(c)
x 2
lim f  x  =
x 2
lim f  x  
x 1
4. Determine each limit.
(a) lim
x0
1

x2
(b) lim
x 4
x
x4
4
(c)
x  3x
x2  9
2
lim e x ln  x  1
(d)
x 1
(e)
lim sin x
x

lim
x 3
2
2.3 Techniques of limits
1. Limits using the properties:
(a)
x   1
2. Let
(a)


lim x 3  2 x 2  6 
(b)
lim f ( x)  2, lim g  x   3, and
x 1
x 1
(b) lim
x 1
x 1
x 1
x4

x  cos x 
2
lim h  x   4 and find each limit:
x 1
lim  2 f  x   3h  x 
lim
f  x h  x
(c)
lim
2 x4 g  x 
x 1
h  x
3. evaluating limits directly.
(a)




lim x 3  x 2  1  x  2, note in this case all that is required is to substitute x = 1 into the expression x 3  x 2  1  x  2.
x   1
When is it okay to just substitute directly into the limit expression? It is okay to substitute into the limit expression when one of the following
forms does not occur:
0
, indeterminate
0
I.
II.
 3  3
(b) lim
x   5
(c)
#
,
0
undefined
tan x
0
x    0 sec 3 x
(d)
lim
4. Limits by factoring: note direct substitution yields
lim 2 sin x  1
x   0
0
.
0
Evaluate:
(a)
x2  2
lim
x 2
x 2

(b) lim
xk
x2  k 2

k 4  x4
(t 2  3)2  9
(c) lim

t  0
t2
x6 1

(d) lim
x   1 x  1
5. Limits by rationalization:
2 x2

x
(a) lim
x   0
(b) lim
x  2
4 x  2
x2
2.5 Continuity
A function f is continuous at a real number
i.
f  a  is defined
ii.
lim f  x  exists
iii.
lim f  x   f  a 
a if the following conditions are met.
x a
x a
A curve is continuous if the curve can be drawn without lifting the pencil. Continuous means, the function has no holes, asymptotes, or jumps.
A curve that is continuous is said to be a smooth curve. A curve that’s not continuous is discontinuous.
5
1. Determine the intervals where the following functions are continuous.
(a)
(b)
6
sx =
4
6
sx =
x
sx =
4
x2+1
2
4
2
2
-2
2
-2
2
-2
-2
-2
-4
-4
-4
-6
-6
-6
2. Which of the functions is continuous at
f ( x) 
6
4
The point (0, 1) is not on the graph.
(a)
x3-1
x-1
x
2
-2
(c)
1
x
(b)
The point (2, 3) is not on the graph.
x  a  5 (a is just a random number)? Explain.
f ( x) 
1
5 x
(c)
 x 2  25
, x5

f ( x)   x  5
10,
x5
3. Determine if the function is continuous at the given value of a .
(a)

x2 , x  1
f  x  
 x  2, x  1
a 1
(b)

- x3  3, x  0
f  x  
x, x  0


a2
Functions with holes (only) can be made continuous by using the conditions of continuity at a point. When function can be patched up, the
hole is said to be a removable discontinuity, otherwise it is a nonremovable discontinuity. Asymptotes, cannot be removed, therefore they
are a nonremovable discontinuity. Only factors that cancel can be removed.
4. Redefine the functions, if possible, so that they are continuous. State the types of discontinuities.
x2
(a) g ( x )  2
x  4 x  12
(d)
4 x
=
f ( x) 
16  x
(b)
4x  1
f ( x) 
2x  3
m 2  2m  3,
(e) r ( m)  
2
 m  2m  1,
(c)
x2 1
g ( x)  3
x  x 2  2x  2
m0
m0
5. Find the limits if they exist.
(a)
h3  27
lim
=
h 3 h  3
(b)
lim
x    3
x3
x3
=
( x  2) 2  4
(c) lim
=
x 0
x
6
6. Determine the value of a, b, and c so that each function is continuous.
(a)
 x 2  bx, x  1

1 x  2
(b) f  x   ax,
 ax3  b, x  2

ax  4, x  1

f  x   2
 x  2a ,
x 1
7. Find the value of the a and b so that the function is continuous for all x.
ax,

(a) f ( x)  
2
2ax  x,
x 1
x 1
(b)
xe
a ln x,
f ( x)  
ax  4,
xe
ax  8,
x  3

 2
3 x  3
(c) f ( x )   ax  4,

bx  8,
x3

2.6 Limits at Infinity
1. Use the graphs to write the horizontal asympotes using limits.
(a)
(b)
(c)
2. Evaluate:
x5  1
x  2 x  6
(b)
lim
sin 2 x
x2  4
(e)
lim
(a)
lim
(d)
lim
x 
A horizontal asymptote, y  a , where
a  bx 4
x  cx 4  8 x 2
x 
3  5x
4 x  12
2
1 x
x  3 x  7 x  4
(c)
lim
(f)
lim 2  e x
x 
a is a real number, is lim f ( x)  a .
x 
3. Find the horizontal asymptote of each function if any.
(a)
(d)
2 x2
4  x2
x 1
x 1
(b)
x
3
x  6x2  x
(e)
1
2  et
(c)
100e2t  t
2  e 2t
(f)
x4  1
x2  1
2
7
2.7 Derivatives and Rates of Change
f ( x  h)  f ( x )
 f ( x)  the first derivative at any point along a differentiable curve.
h
0. Slope of tangent line is m  lim
h  0
Derivation:
Notation for the derivative:
I.
f  x,
f
II. f ( x)  lim
h   0
f   c   lim
x  c
prime of
x
II . y ,
y
prime
III .
dy
, derivative of
dx
y
with respect to
f ( x  h)  f ( x )
can also be rewritten by letting c  x  h  h  c  x, where x    c.
h
f ( x )  f (c )
is the slope of a differentiable curve at a specific point  c, f  c   .
xc
1. Find the derivative (the slope of the tangent line) for each curve using the definition of the derivative.
f ( x)  ax 2 , a  0.
(a)
(b)
f ( x)  2 x .
2. Find the first derivative of each function with respect to the specified variable, by using the limit definition of the derivative.
(a)
y  vx  w, x
(c)
g ( x)  x 3  6 x
(b)
y (t ) 
1
,t
t
(d)
f ( x)  x 2  4
(c)
 2( x  h)  ( x  h) 3  2 x  x 3
h   0
h
3. Evaluate these and remember that cos (a+b) = cos a cos b – sin a sin b
(a)
( x  h) 2  x 2
h   0
h
lim
cos( x  h)  cos x
h   0
h
(b)  2 lim
lim
4. What do the following limits represent? In other words, what derivatives are being evaluated? What is the value of the limit?
(a)
2


sin   h  
4
 2
lim
h   0
h
(b)
(3  h) 3  2(3  h) 2  9
h   0
h
lim
x
5. Find the indicated derivative. In the book the instructions say to find f ( a ) . Use the alternate form of the derivative . It is the form that
is most useful when finding the derivative at a particular value.
(a) Find f ( 2) for
f ( x)  x 2  4
(b) Find g (1) for g  x  
x.
6. Find the equation of the tangent line to the curve at the given point. Confirm using your graphing calculators.
(a) y  f  x  
1
,
x
1, 1
(b)
7. Find the equation of the line tangent to the curve
y  3t  t 2 ,
 1,  4
y  2 x 2  8 x  7 which is parallel to the line y  4 x  2  0 .
8 For what value(s) of x, will the curves x  4 x  6 and  3 x  10 x  6 have the same rate of change.
2
2
9. Between what consecutive points is the rate of change the least? The greatest? Zero? The graph given is the graph of f ( x ) .
10
B
8
6
G
C
4
A
D
F
2
-5
E
5
-2
10. Find the average rate of the given functions and the instantaneous velocity at the given t values.
Ave. rate =average velocity 
(a)
f (t )  2t  t 3 , [0, 1]
Instantaneous rate when t = 1.
f (b)  f (a)
ba
(b)
h(t )  sin t ,
Instantaneous rate when t =
 
 tell you about the function? Assume units are meters per second.
2
What does v (1) and v

.
2
  
6 , 2


8
9
11. The position function of vehicle in motion is sketched below. Draw the velocity function. When is the car at rest? When is the car
moving to the right and to the left?
12
3
2
1
-2
t1
2
t2
4
-1
.
a) Where is the particle traveling faster, at t1 or t 2 ?
b) Is the particle speeding up or slowing down on the interval [ t1 or t 2 ] ?
13. Find the equation of the tangent line to the graph of
y  g  x  , where g  2  2 and
g   2  3 .
10
14. Sketch the velocity of function of a particle whose position graph is sketched below. When is the particle moving left? Right?
2.8 The Derivative as a Function
1. Find the equation of the tangent line to
y  4  x 2 at the point where x  1 .
2. The following graphs (a) – (d) are the graphs of functions. The graphs (i) – (iv) are the graphs of the derivative of each function. Match the
graph to its derivative.
(a)
(b)
(c)
(d)
6
4
4
4
3
2
4
2
2
-5
5
1
2
5
-2
2
4
6
-1
5
-2
-4
(i)
(ii)
(iii)
4
(iv)
2
4
2
1
2
2
1
2
2
4
-5
5
-5
5
-1
-1
-2
-2
-4
-4
-2
-2
-3
4
11
3. Sketch the graph of each derivative given the function.
(a)
(b)
(c)
Remember the derivative does not exist at sharp turns, cusps, or jumps. If a function is continuous does it imply that it is differentiable?
2
.
x 1
(a) Find the domain of f  x  .
4. Given y  f  x  
(b) Determine when
5. Given
f   x   0 and f   x   0 , write the answer in interval notation. What is the meaning of the intervals?
y  x3  x
(a) Find the domain of
(b) Determine when
f  x .
f   x   0 and f   x   0 , write the answer in interval notation. What is the meaning of the intervals?
6. Determine the intervals where the derivate is positive and negative and sketch its graph on the same plane as the functions below.
(a)
(b)
(c)
(d)
12
7. Two graphs are given f ( x) and f ( x). which one is which? Explain how you made your choices.
(a)
(b)
10
10
5
5
10
-5
-10
10
-10
-5
-15
8. In each graph determine whether the derivative is increasing, decreasing, or remains constant.
(a)
(b)
(c)
(d)