Transcript MTH 251 – Differential Calculus

MTH 251 – Differential Calculus
Chapter 2 Review
Limits and Continuity
Copyright © 2010 by Ron Wallace, all rights reserved.
Calculating Limits - 1
lim f ( x )
xc
• Polynomial: f(c)
• Rational: f(c) if there is
no division by 0
• Radical: f(c) if there are
no even roots of
negatives.
• Exponential: f(c)
• Logarithmetic: f(c) if
there a no logs of nonpositives
• Trigonometric: f(c)
except where the trig
functions are undefined.
• Sums, Differences,
Products, Quotients
(except division by zero), and
Compositions of these.
Calculating Limits - 2
f ( x)
lim
xc g ( x)
m
f ( x)
lim
xc g ( x)
m
n
0
f ( x)
lim
xc g ( x)
f ( x)
lim
xc g ( x)
Note: Assume m  0 & n  0
0
n
0
0
Calculating Limits - 2
x 5
lim
x 2 x  2
x 9
lim 2
x 3 x  3
5x  1
x5
lim
x 5 x  5
2
lim
x 1
 x 1
2
2
Calculating Limits - 2
f ( x)
lim
xc g ( x)
0
0
• Algebraic equivalences.
 Remove (x – c) if it is a common factor.
 Rationalize the numerator or denominator
Calculating Limits - 2
f ( x)
lim
xc g ( x)
0
0
• Algebraic equivalences.
 Remove (x – c) if it is a common factor.
 Rationalize the numerator or denominator
x  3x  10
lim
2
x 2
x 4
2
Calculating Limits - 2
f ( x)
lim
xc g ( x)
0
0
• Algebraic equivalences.
 Remove (x – c) if it is a common factor.
 Rationalize the numerator or denominator
x 5 3
lim
x 4
x4
Calculating Limits - 3
sin mx
lim
x 0
mx
1
lim
x  x
ax  
lim n
x  bx  
m
mn
mn
mn
Calculating Limits - 4
f ( x)
lim
xc g ( x)

f ( x)
lim
xc g ( x)

n
0
f ( x)
lim
xc g ( x)
m
f ( x)
lim
xc g ( x)
0
Note: Assume m  0 & n  0


Calculating Limits - 5
lim f ( x )
x c
• Left-Hand Limit
• Only need to
consider x < c
lim f ( x )
x c
• Right-Hand Limit
• Only need to
consider x > c
Calculating Limits – 3, 4, & 5
3x  11
lim 2
x  5 x  7 x
2
lim
x 5
7x
x 5
sin 2 x
lim
x  0 x cos x
cos 3 x
lim

tan x
x
2
When a limit DNE
lim f ( x )  DNE
x c
• f(x) is not defined around c
• Jump
• Usually a piecewise function
• Oscillation
• Usually involves sine or cosine
• Increase/Decrease without bound
• Vertical asymptotes
• May be different on left and right
Proving Limits
• Definition
lim f ( x )  L
x c
f(x)
if, for every  > 0,
there exists a  > 0
such that …
0  x  c    f ( x)  L  
L+
L
if,    0    0 
0  x  c    f ( x)  L  
L-
c- c c+
You MUST be able to state this & draw the diagram w/ labels.
Proving Limits
• Process …
 Begin with f ( x)  L  
• i.e.
  f ( x)  L  
lim f ( x )  L
x c
 may or may
not be given
 Manipulate to get … a  x  c  b
 Determine 
  min  a , b
note: a  0 & b  0
Proving Limits
• Prove that …
using
lim x  5  3
3
x 2
  0.01
Round calculations to 5 decimal places.
Continuity
• f(x) is continuous at x = c if and only if …
 When c is an interior point of the domain and
lim f ( x )  f ( c )
xc
 When c is a left endpoint of the domain and
lim f ( x )  f (c )
x c
 When c is a right endpoint of the domain and
lim f ( x )  f ( c)
x c
NOTE: You MUST be able to state this definition.
Continuous Functions
• A function that is continuous for all
values of its domain.
• All of the “elementary functions” are
continuous functions.
 polynomials, rationals, radicals,
exponentials, logarithms, absolute values,
trigonometric, and combinations of these
– note: Consider the domains
Common Points of
Discontinuity
• f(c) is not defined




division by zero
square roots of negatives
asymptotes of trigonometric functions
logarithms of non-positives
• The limit as x  c DNE
 oscillations
• f(c) does not equal the limit as x  c
 piece-wise functions where the endpoints of the
pieces don’t connect
Discontinuity Examples
• Where are these functions NOT continuous?
f ( x)  x  6 x  5
3
x5
f ( x)  2
x  25
 x 2 +1, x  0
f ( x)  
 2 x  1, x  0
4
x  2  1