LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions...
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Transcript of LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions...
![Page 1: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/1.jpg)
LIMITSOF
FUNCTIONS
![Page 2: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/2.jpg)
CONTINUITY
![Page 3: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/3.jpg)
Definition 1.5.1 (p. 110)
If one or more of the above conditions fails to hold at C the function is said to be discontinuous.
DEFINITION: CONTINUITY OF A FUNCTION
![Page 4: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/4.jpg)
Theorem 1.5.3 (p. 113)
![Page 5: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/5.jpg)
Question 8
EXAMPLE
Solution:
2 2 36
3
x xx xf xx
3x
2
2 3
x
f x x where x
1. Given the function f defined as ,draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.
2 6
3x xf x
x
![Page 6: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/6.jpg)
y
x
Test for continuity: at x=3
1.f(3) is not defined; since the first condition is not satisfied then f is discontinuous at x=3.
![Page 7: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/7.jpg)
Question 8
2. Given the function f defined as
draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.
EXAMPLE
if
if
2 6 33
2 3
x x xf x xx
![Page 8: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/8.jpg)
y
x
Solution:
![Page 9: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/9.jpg)
3x at ousdiscontinu is f therefore
)3(f)x(flim but exists itlim The
2)3(f and 5)x(flim .3523
2xlim 3x
2x3xlim
3x6xxlim)x(flim .2
2 to equal is and defined is 3f .1:continuity for Test
3x
3x
3x
3x
2
3x3x
![Page 10: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/10.jpg)
Question 8Question 8
EXAMPLE
3. Given the function f defined as ,
draw a sketch of the graph of f, then by observing where there are breaks in the graph, determine the values of the independent variable at which the function is discontinuous and why each is discontinuous.
if
if 2
1 0
2 0
xf x x
x
![Page 11: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/11.jpg)
20
20
1 1lim0
1 1lim0
0
x
x
x
x
x is a VA
VA a is 0x
0x if x1)x(f
:Graph
2
2
2
1lim 0
1lim 0
0
x
x
x
x
y is a HA
HA
0x at ousdiscontinu is f then satisfied not is
condition ondsec the Since
exists not does lim .2defined;2)0(f .1
:continuity for Test
0x
![Page 12: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/12.jpg)
Question 8Question 8
Solution:y
x
1
-1 1
![Page 13: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/13.jpg)
Figure 1.5.1 (p. 110)
The figure above illustrates the function not defined at x=c, which violates the first condition.
The figure above illustrates that the limit coming from the right and left both existbut are not equal, thus the two sided limit does not exist which violates the second condition. This kind of discontinuity is calledjump discontinuity.
![Page 14: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/14.jpg)
Figure 1.51 (p. 110)
The figure above illustrates that the limit coming from the right and left of c are both , thus the two sided limit does not exist which violates the second condition. This kind of discontinuity is called infinite discontinuity.
The figure above illustrates the function defined at c and that the limit coming from the right and left of c both exist thus the two sided limit exist. But which violates the third condition. This kind of discontinuity is calledremovable discontinuity.
)x(flim)c(fcx
![Page 15: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/15.jpg)
2
y
x
)x(fy
2x4x)x(f given
; 2x at continuous is function the whether eminDeter2
2x at continuousdis hence and2x at undefined is )x(f thus
2x but ;422)2(f
2x)x(f2x
2x2x)x(f
4
![Page 16: LIMITS OF FUNCTIONS. CONTINUITY Definition 1.5.1 (p. 110) If one or more of the above conditions fails to hold at C the function is said to be discontinuous.](https://reader036.fdocuments.net/reader036/viewer/2022081521/5a4d1b6e7f8b9ab0599b44d6/html5/thumbnails/16.jpg)
2
4
y
x
)x(gy
3
2x ,3
2x ,2x4x
)x(g given
; 2x at continuous is function the whether eminDeter2
2x at continuousdis hence
)2(g)x(glim the cesin3)2(g
4)x(glim thus
4)x(glim)x(glim2x)x(g
2x2x2x)x(g
2x
2x
2x2x
Removable Discontinuity
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2x ,4
2x ,2x4x
)x(h given
; 2x at continuous is function the whether eminDeter2
2x at continuous hence
)2(h)x(hlim the cesin4)2(h
4)x(hlim thus
4)x(hlim)x(hlim2x)x(h
2x2x2x)x(h
2x
2x
2x2x
2
4
y
x