Chapter 2 Section 2
Continiuty
® y = f(x) whose graph can be sketched over any interval
of its domain with one continuous motion is a continuous function.
This is continuous on the interval from [a,b].
This is discontinuous at 1 and 2.
Then we say that 1 and 2 are points of discontinuity.
The limit does not exist at 1 because left and right limits are not the same.
Algebraic Properties of Continuous Functions:
The sum of two continuous functions is also continuous.
Removing discontinuities:
Have to simplify algebraically:
Ex: x+6 = x+6
x+7x+6 (x+6)(x+1)
discontinuous at -1 because make the denominator zero.
Composition of Continuous
Functions:
All compositions of continuous functions are continuous.
Maximum and Minimum Theorem:
Examples:
Give points of discontinuity:
| 1. y = | x+4 | 2. y = | cos x |
| x2-2x-24 | x | ||
| 3. y = | 5 | 4. y = | Ö3x+2 |
| x+6 | |||
| 5.
lim x®0 |
tan x | ||
Scroll down for answers when done.
Answers for Examples:
Give points of discontinuity:
| 1. y = | x+4 x+4 | 2. y = | cos x |
| x2-2x-24
(x-6)(x+4) discont. at x = 6 |
x discont. at x = 0 |
||
| 3. y = | 5 | 4. y = | Ö3x+2
3x+2<0 3x<-2 x<-2/3 discont. at -2/3 |
| x+7 discont. at x = -7 |
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| 5.
lim x®0 |
tan x
= lim sinx
= sin0 = 0 = 0 x®0 cosx cos0 1 |
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