Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Lecture 6–7: LimitsThomas’ Calculus Sections 2.2, 2.4

Oscar F Bandtlow

School of Mathematical SciencesQueen Mary University of London

Calculus I — Week 2

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Limits

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Motivating example

Let

f (x) =x2 − 1

x − 1(x 6= 1) .

How does the function f behave near the x-value 1?

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Informal definition of a limit

Definition

Let f be a function defined in an open interval containing thepoint c (apart possibly from the point c itself). We say f (x)approaches the limit value L as x approaches c , if f (x) is‘arbitrarily close’ to L whenever x is ‘sufficiently close’ to c . Ifthis is the case we write

limx→c

f (x) = L .

Example

f (x) = k limx→c

f (x) =

g(x) = x limx→c

g(x) =

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Limits need not exist

For none of the following functions the limit at 0 exists:

f (x) =

{0, x < 0

1, x ≥ 0

g(x) =

{1x , x 6= 0

0, x = 0

h(x) =

{0, x ≤ 0

sin( 1x ), x > 0

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Limit laws

Theorem

Let f and g denote functions and assume that the limits

limx→c

f (x) = L and limx→c

g(x) = M

exist. Then the following limits exist as well:

1 limx→c

(f (x) + g(x)) = L + M

2 limx→c

(f (x)− g(x)) = L−M

3 limx→c

(k · f (x)) = k · L, (k ∈ R)

4 limx→c

(f (x) · g(x)) = L ·M

5 limx→c

f (x)

g(x)=

L

M, provided M 6= 0

6 limx→c(f (x))r/s = Lr/s , (r , s integers, s 6= 0, and Lr/s

real)

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Revisiting the motivating example

Let

f (x) =x2 − 1

x − 1.

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Something to remember

Which of these limits exist?

limx→1

f (x) limx→1

g(x) limx→1

h(x)

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Another example

Let f (x) =√

4x2 − 3. Using the Limit Laws find

limx→3

f (x) .

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Another example continued

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Yet another example

Let

g(x) =

√x2 + 100− 10

x2.

Findlimx→0

g(x) .

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Yet another example continued

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

The Sandwich Theorem

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

The Sandwich Theorem

Theorem (Sandwich Theorem)

Suppose that f , g and h are functions defined on an openinterval I containing the point c (but the functions need not bedefined at the point c itself). Suppose further that

g(x) ≤ f (x) ≤ h(x) for all x ∈ I \ {c}and that

limx→c

g(x) = limx→c

h(x) = L .

Thenlimx→c

f (x) = L .

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Example: limit of sine at 0

Using the Sandwich Theorem, show that

limθ→0

sin(θ) = 0 .

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Example: limit of cosine at 0

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Example: another limit

Using the Sandwich Theorem, find

limθ→0

sin(θ)

θ.

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

One-sided limits

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

One-sided limits

Definition

Let f be a function defined on an interval (c , b). If f (x) getsarbitrarily close to L as x approaches c from within (c , b), thenwe say f has a right-hand limit L at c and we write

limx→c+

f (x) = L.

Definition

Let f be a function defined on an interval (a, c). If f (x) getsarbitrarily close to L as x approaches c from within (a, c), thenwe say f has a left-hand limit L at c and we write

limx→c−

f (x) = L.

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Example

Considerf (x) =

x

|x |(x 6= 0)

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Relation between limits and one-sided limits

Theorem

Suppose f is a function defined on an open interval containingc , except perhaps at c itself. Then f has a limit as xapproaches c if and only if it has left-hand and right-handlimits at c and the limits are equal:

limx→c−

f (x) = L and limx→c+

f (x) = L if and only if limx→c

f (x) = L .

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Addenda

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Addenda

Lecture 6–7

OF Bandtlow

Limits

Limit of a functionand Limit Laws

The SandwichTheorem

One-sided limits

Addenda