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Aim: What is the limit?. Do Now: Evaluate the functions. 1. f ( x ) = 3 x – 2, when x = 2. 2. , when x = 1. 3. when x = 0. 1. f (2) = 6 – 2 = 4. 2. f (1) is undefined

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Aim: What is the limit?

Do Now: Evaluate the functions

1. f(x) = 3x – 2, when x = 2

2. , when x = 1

3. when x = 0


1. f(2) = 6 – 2 = 4

2. f(1) is undefined

We can not evaluate f(1) by direct substitution, but we can estimate f(x) reaches 2 when x approaches 1

3. f(x) getting larger and larger when x approaches to 0

Definition of Limit

If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L. This is written as


The limit of f(x) as x approaches 2 can be obtained by direct substitution.

The limit of f(x) as x approaches 1 can not be obtained by direct substitution, but we can estimate or by factoring.

In this case, the limit of f(x) does not exist, since f(x) reaches infinity as x approaches 0


We can conclude that the limit of a function f(x) as x approaches c sometimes exists and sometimes not, depending on the function and the value of c

There are different methods to evaluate the limit of a function if the limit exist.

  • Direct substitution

  • Cancellation

  • Rationalize

The following examples will illustrate those three situations


Direct substitution

Cancellation

Rationalization

By direct substitution we get



When does a limit does not exist? the

1. When the left and right limit are not equal

y

.

1

x

y

2. Unbounded behavior

x



Evaluate the limit the

1.

2.

4

1

3.

4. If f(x) = x2 – 3x

find

2x – 3


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