Consider We want to find , but the quotient rule is not applicable! Factorize and hence

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Lecture2| 1 Techniques to find limits

1. Algebraic manipulation

Fact. If for all , except possibly at , and exists, then the limit of as

also exists and

Consider

We want to find , but the quotient rule is not applicable!

Factorize and hence

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Lecture2| 2 The common terms cancelled and

true for all .

Now by Rule 5 and the fact above,

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Lecture2| 3 EXAMPLE (Factorization). Find the limit

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Lecture2| 6 Question Use factorization to find the limit

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Lecture2| 7 EXAMPLE (Simplify). Find the limit

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Lecture2| 8 2. Conjugation

We know . Setting

we get Fact.

Variants

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Lecture2| 9 EXAMPLE. Find the limit

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Lecture2| 11 Question Find the limit

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Lecture2| 12 Question Use both factorization and conjugation to find the limit

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Lecture2| 13 3. Left- and right-hand limits

For functions defined by two or more formulas, e.g.

finding the limit at a break-point requires the left- and right-hand limits.

Theorem. exists if

1. , exist, and

2. .

Then the limit is the common value.

Fact. All the limit laws can be applied.

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Lecture2| 14 EXAMPLE. Let

1. Find the left- and right-hand limits at . 2. Determine whether the limit exists at .

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Lecture2| 15 EXAMPLE. Find the limit (if it exists)

Note. is defined by two formulas

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Lecture2| 16 EXAMPLE. Given that the function

has the limit at , find .

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Lecture2| 17 Question For the function

find all that has .

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Lecture2| 18 The Squeeze Theorem. Consider the limit

We know that and , so

By limit laws, we have

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Lecture2| 19 This means and approaches as , so the function which is squeezed between

and must have the same behavior.

Rule (The Squeeze Theorem). If

and

then

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Lecture2| 21 1.4. Infinite Limits

Consider

as approaches .

From the table, does not approach any

number as or , so both

and do not exist.

 decreases without bound as

 increases without bound as These special behaviors are denoted

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Lecture2| 22

Note. are symbols (not numbers) that represent the behaviors of increasing without bound and decreasing without bound, resp.

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Lecture2| 23 Note. When as approaches from any direction, one often draws a vertical line at

in the graph .

This line is called a “vertical asymptote”.

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Lecture2| 24 EXAMPLE.

Vertical asymptote:

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Lecture2| 25 EXAMPLE.

Vertical asymptote:

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Lecture2| 26

Question Discuss the behavior of

as approaches . Find all vertical asymptotes of

the graph .

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Lecture2| 27

How to spot infinite limits?

For the limit , if we calculate

so, quotient rule cannot be applied!

Since , . So is a negative

number closes to zero. E.g. , then

This explains why as .

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Lecture2| 28

Rule (Infinite limits). If 1.

2. and as ,

then

Variants.

(a) and , then

(b) and , then

(c) and , then .

(d) The rule can be applied to right-hand limit and the full limit.

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Lecture2| 29

EXAMPLE. Find the following limits 1.

2.

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Lecture2| 30 3.

4.

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Lecture2| 31 EXAMPLE. Discuss the limits

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Lecture2| 34 Question Determine whether the limit is or

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Lecture2| 35 Summary. can be one of the following

forms (from the quotient rule):

1. and (the limit exists)

2. and (the limit does not exist, ) 3. (do factorization, conjugation, etc. first)

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Lecture2| 36 Question Let

Find the following limits 1.

2.

3.

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Lecture2| 37 EXAMPLE. Find all vertical asymptotes (vertical line , where ) for the graph of