![Page 1: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/1.jpg)
Introduction to Limits (2.1 & 2.2)
Xiannan Li
Kansas State University
January 22nd, 2017
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 2: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/2.jpg)
Secant lines
1 Recall that the essential idea in differential calculus is nicefunctions are well approximated by lines near a point (andthe rate of change corresponds to the slope of that line).
2 Given a function f(x), a secant line is simply a line passingthrough two points of the form (x1, f(x1)) and (x2, f(x2)).
3 As x2 gets closer and closer to x1, this secant line becomesa better and better approximation to the function near x1.In the ”limit” process, these secant lines approach the”tangent line” at x1.
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 3: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/3.jpg)
Example: how do we find the average velocity?
Let s(t) denote the car’s position (in feet) at time t (in seconds).
Average Velocity from t1 to t2 =∆s
∆t=
s(t2)− s(t1)
t2 − t1
This is the same as the slope of the secant line from (t1, s(t1))to (t2, s(t2)).
What about instantaneous velocity?(think of the reading on your speedometer)
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 4: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/4.jpg)
The average velocity between t = 1 sec and t = 2 sec is
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 5: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/5.jpg)
The average velocity between t = 1 sec and t = 1.5 sec is 4.75 ftsec .
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 6: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/6.jpg)
The average velocity between t = 1 sec and t = 1.25 sec is 3.8125 ftsec .
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 7: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/7.jpg)
The average velocity between t = 1 sec and t = 1.05 sec is 3.1525 ftsec .
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 8: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/8.jpg)
The average velocity between t = 1 sec and t = 1.01 sec is 3.0301 ftsec .
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 9: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/9.jpg)
The secant lines approach the tangent line to the curve at t = 1 sec.
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 10: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/10.jpg)
As the time intervals get smaller and smaller, the secant linesare getting closer and closer to the tangent line to the curve att = 1 sec, and the slopes of the secant lines are getting closerand closer to 3 ft/sec.
By taking smaller and smaller time intervals, the averagevelocity is starting to become close to the instantaneous velocityat t = 1 sec. (Think of this as what your speedometer says.)
In fact, with calculus, one can show the instantaneous velocitywhen t = 1 sec is 3 ft/sec.
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 11: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/11.jpg)
In the previous example, we have already seen the rough shapeof a ”limit” process, as one point gets closer and closer toanother.But what exactly are limits and how do we calculate themprecisely?
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 12: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/12.jpg)
Definition (left-hand limits)
We write limx→a−
f(x) = L if we can make f(x) arbitrarily close to
L by taking x sufficiently close to a and strictly less than a.
limx→2−
f(x) = limx→3−
f(x) =
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 13: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/13.jpg)
Definition (right-hand limits)
We write limx→a+
f(x) = L if we can make f(x) arbitrarily close to
L by taking x sufficiently close to a and strictly greater than a.
limx→2+
f(x) = limx→3+
f(x) =
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 14: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/14.jpg)
Definition (limits)
We write limx→a
f(x) = L if we can make f(x) arbitrarily close to
L by taking x sufficiently close to a but not equal to a.
limx→a
f(x) = L if and only if
limx→a−
f(x) = L
limx→a+
f(x) = L
limx→1
f(x) = limx→2
f(x) = limx→3
f(x) =
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 15: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/15.jpg)
Definition (limits)
We write limx→a
f(x) = L if we can make f(x) arbitrarily close to
L by taking x sufficiently close to a but not equal to a.
limx→a
f(x) = L if and only if
limx→a−
f(x) = L
limx→a+
f(x) = L
limx→2−
f(x) = limx→2+
f(x) = limx→2
f(x)
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 16: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/16.jpg)
Definition (limits)
We write limx→a
f(x) = L if we can make f(x) arbitrarily close to
L by taking x sufficiently close to a but not equal to a.
limx→a
f(x) = L if and only if
limx→a−
f(x) = L
limx→a+
f(x) = L
limx→0−
f(x), limx→0+
f(x), and limx→0
f(x) do not exist
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 17: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/17.jpg)
Numerically, we can check for x = − 1π/2 ,−
15π/2 ,−
19π/2 , ... that
sin1
x=
Also, for x = − 13π/2 ,−
17π/2 ,−
111π/2 , ...,
sin1
x=
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 18: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/18.jpg)
Let g(x) =x2 + 2x− 3
x− 1. Find lim
x→1g(x).
x g(x)
.5 3.5
.95 3.95
.999 3.999
1 undefined
1.001 4.001
1.05 4.05
1.5 4.5
limx→1
g(x) = (even though g(1) is undefined)
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 19: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/19.jpg)
Intuitive Idea of Limits
limx→2
f(x) = 1.
It doesn’t matter what f(2) is or even if it is defined.As x gets “close” to 2, f(x) gets close to 1.
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 20: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/20.jpg)
Looking back at g(x) = x2+2x−3x−1 . How do we precisely evaluate
limx→1 g(x)?
Note that x2 + 2x− 3 =
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 21: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/21.jpg)
Definition (vertical asymptotes)
We say that the line x = a is a vertical asymptote of y = f(x)provided that one of the following holds:
limx→a
f(x) = ±∞
limx→a−
f(x) = ±∞
limx→a+
f(x) = ±∞
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 22: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/22.jpg)
x = 2 is a vertical asymptote
limx→2−
1
x− 2= lim
x→2+
1
x− 2= lim
x→2
1
x− 2
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)
![Page 23: Introduction to Limits (2.1 & 2.2) - Kansas State University · Introduction to Limits (2.1 & 2.2) Xiannan Li Kansas State University January 22nd, 2017 Math 220 { Lecture 2 Introduction](https://reader031.vdocument.in/reader031/viewer/2022021418/5acb09c07f8b9a6b578e1e0a/html5/thumbnails/23.jpg)
LOOKING AHEAD
How do we find the slope of thetangent line to y = f(x) at x = a?
limt→a
f(t)− f(a)
t− a
We find the average rate of change of f(x) on the interval froma to t, and we look at the limit of this quantity as t approachesa, which means the interval’s length is shrinking to zero.
This is a powerful tool called the derivative.
Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)