![Page 1: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/1.jpg)
Determine if
)(lim2xg
x
exists for the functions in the following graphs
![Page 2: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/2.jpg)
![Page 3: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/3.jpg)
![Page 4: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/4.jpg)
![Page 5: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/5.jpg)
Infinity and Limits
Consider the function f(x) whose graph is
As x increases, the value of f(x) approaches 2.
![Page 6: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/6.jpg)
In this case, we say that the limit of f(x) as x approaches infinity is 2.
We express this using limit notation as
2)(lim
xfx
![Page 7: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/7.jpg)
Similarly, if we examine the following graph, we will note that as x grows large in the negative direction, the value of f(x) approaches 0.
![Page 8: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/8.jpg)
We express this using limit notation as
0)(lim
xfx
![Page 9: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/9.jpg)
Example
1
1lim
2xx
![Page 10: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/10.jpg)
1.5 Differentiability and Continuity
![Page 11: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/11.jpg)
If a is a constant, we say that f(x) is differentiable at x = a if we can evaluate the following limit to determine f 0(a).
h
afhafaf
h
)()(lim)('
0
Conversely, if this limit does not exist, then f(x) is nondifferentiable at x = a.
![Page 12: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/12.jpg)
There are many geometric representations of f(x) for functions that are nondifferentiable at x = a.
These can result if f(x) has no tangent line at x = a, or if f(x) has a vertical tangent line at x = a.
![Page 13: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/13.jpg)
![Page 14: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/14.jpg)
![Page 15: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/15.jpg)
![Page 16: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/16.jpg)
![Page 17: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/17.jpg)
A railroad company charges $10 per mile to haul a boxcar up to 200 miles and $8 per mile for each mile exceeding 200. In addition, the railroad charges a $1000 handling charge per boxcar.
Graph the cost of sending a boxcar x miles.
![Page 18: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/18.jpg)
If x is at most 200 miles, then the cost C(x) is given by:
C(x) = 1000 + 10x dollars
If x exceeds 200 miles, then the cost will beC(x) = 3000 + 8(x – 200) = 1400 + 8x
So the function C(x) is given by
200,81400
2000,101000)(
xx
xxxC
![Page 19: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/19.jpg)
The graph of C(x) is
![Page 20: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/20.jpg)
Continuity
![Page 21: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/21.jpg)
Continuity is closely related to the concept of differentiability.
We say that a function is continuous at x = a if its graph has no breaks or gaps as it passes through the point (a, f(a)).
If a function f(x) is continuous at x = a, it should be possible to sketch its graph without lifting the pencil from the paper at the point (a, f(a)).
![Page 22: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/22.jpg)
The following graphs depict functions that are not continuous.
![Page 23: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/23.jpg)
![Page 24: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/24.jpg)
![Page 25: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/25.jpg)
If f(x) is differentiable at x = a, then f(x) is continuous at x = a.
So, a function that is differentiable at x = a will be continuous at x = a.
Note however, it is still possible for a function to be continuous at x = a, but not differentiable.
![Page 26: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/26.jpg)
Expressing continuity in terms of limits, we have the Limit Definition of Continuity
A function f(x) is continuous at x = a provided the following limit relation holds:
)()(lim afxfax
![Page 27: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/27.jpg)
)()(lim afxfax
In order for this to hold, three conditions must be fulfilled.
1. f(x) must be defined at x = a
)(lim xfax2. must exist
3. The limit )(lim xfax
must have the value f(a)
![Page 28: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/28.jpg)
Which of the graphs represent continuous functions?
![Page 29: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/29.jpg)
![Page 30: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/30.jpg)
![Page 31: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/31.jpg)
![Page 32: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/32.jpg)
![Page 33: Determine if exists for the functions in the following graphs](https://reader036.vdocument.in/reader036/viewer/2022062314/56649f2b5503460f94c45c98/html5/thumbnails/33.jpg)