What is a limit ?

When does a limit exist?

Continuity Discontinuity

Types of discontinuity

What is a limit?A limit of a function is an intended

“height” of a function at a point

Lim f (x) = 4

X 2

When does a limit exist?

When does a limit exist?f (x) = x²

When does a limit exist?

When does a limit exist?

With a plus sign

When does a limit exist?

With a minus sign

In order to have a limit at a point

At that point:

Left limit MUST EQUAL Right limit

Lim f (x)= 1 Lim f (x)= 1

X 4ˉ X 4

Lim f (x)= 1

X 4With no sign

+

Limit exist

Lim f (x)= 1 X 4

Infinite Limits( are not limits)

How to evaluate a limit, in case of algebraic functions?

1. Finding the value of the function at the point (Substitution in the formula of the function) If the function is continuous at the point

2. Factoring (The case 0/0 )

3. The Conjugate Method (The case 0/0 )

Substitution

Factoring

Factoring

The Conjugate MethodWhat is a conjugate?

X-16 = (√x - 4) (√x + 4)

AND(√x + 4) is the conjugate of (√x - 4)respect to X-16

(√x - 4) is the conjugate of (√x + 4)respect to X-16

The Conjugate Method

Continuity

f is continuous at a if:

)()(lim afxfax

Example of a function f which is discontinuous, but continuous from the right.

lim f (x) = 4 = f (2)X2+

Example of a function f which is discontinuous, but continuous from the left.

lim f (x) = 4 = f (2)X2ˉ

Discontinuity

A function is discontinues at a if the limit at a is not equal to the value f (a)

A continuous function on R should have:1.no breaks in the graph2.no holes 3.no jumps

Everywhere Continuous Function

24)(lim

xxf

24)(lim

xxf

2

4

* Since f(2) is also equal to 4; then = f(2)

• The function is continuous at 2.

• And since it’s also continuous at all other point’s in R; then it’s everywhere continuous.

Type of discontinuity

1. Removable discontinuity

2. infinite discontinuity

3. jump discontinuity

Jump discontinuity

7)(lim1

x

xf

4)(lim1

x

xf

existnotdoesxfx 1

)(lim

1

7

4

f is continuous on the intervals (-∞ ,1 ) and [1, ∞ )

Infinite discontinuity

4)(lim3

xfx

)(3

xfLimx

3

4

The function is continuous on the interval (-∞ , 3] and (3, ∞ )

Notice that the function is continuous from the left at 3

Removable discontinuity: the limit of f exists at the point but f is not equal to the value of f at

that point.

4)(lim3

x

xf

4)(lim3

xfx

4)(lim3

xfx

3

Notice that the function is neither continues from the left nor continuous from the right at 3

The function is continuous on the interval (-∞,3) and (3,∞)

4

Let

1

2

3

-2

(1,3)

f is not defined at 1 but exists and equal to 3)(lim1

xfx

}1{;2)( Rxxxf

Removable discontinuity: the limit of f exists at the point but f is not defined at that point.