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)

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)

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)

The average velocity between t = 1 sec and t = 2 sec is

Math 220 – Lecture 2 Introduction to Limits (2.1 & 2.2)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)