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Chapter 1 Limits and Continuity

Outline 5. Limit Laws 6. Continuity

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5. Limit Laws We had formulated the idea that

using tables or graphs.

The value is called the limit and is denoted

However, we need a quick and precise way to calculate the limits.

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Limit Laws Suppose is a constant, are functions such that

Then

1. (Sum)

2. (Subtraction)

3. (Constant multiplication)

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4. (Product)

5. (Quotient)

provided Remark The limit laws are easily seen to be true. For example, if is close to and is closed to , then is close to . So it is intuitively clear that the sum law is true.

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6. (Power)

where 7. (Root)

where If (even), we assume that .

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8. (Special)

and

where If (even) we assume .

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EX Evaluate the limit

Remark Every polynomial

has the direct substitution property at every , i.e.

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EX Evaluate the limit

Remark Every rational function

has the direct substitution property at every such that , i.e.

provided .

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EX Evaluate the limits

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EX (The limit laws are true for one-sided limits) Evaluate the limits

Remark In (3), the function stays positive as , so the root law is true.

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EX (Do some preliminary algebra) Evaluate the limit

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EX Evaluate the limits

Remark

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EX (Absolute value functions) Evaluate the limit

by simplifying the function

as .

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Remark

For finding a limits with , we can assume and close to . For finding a limit with , we can assume and close to .

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EX Show that the limit

does not exist.

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EX (Rationalization) Evaluate the limit

Remark

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EX Find

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Theorem (Squeeze) If when is close to and

then

Proof

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EX Let be a function such that

for close to . Evaluate the limit

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EX Show that

by verifying that

Remark

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6. Continuity Def A function is continuous at if

I.e. the DSP is true for at .

If is not continuous at , we say that

is discontinuous at .

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To be continuous at , all the

following conditions must be true:

Failing to satisfy at least one of the

above conditions implies that the function is discontinuous at .

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EX Determine whether the function

is continuous at ? Find the set of all points where is continuous.

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EX For the following graph of a function, is the function continuous at ?

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EX Explain why the function is discontinuous at the given number ?

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Def A function is continuous from the right at if

and is continuous from the left at if

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Def is continuous on if is cont. at every . is continuous on if is cont. on and it is cont. from the right at . is continuous on if is cont. on and it is cont. from the left at . is continuous on if is cont. on , is cont. from the right at , and is cont. from the from the left at .

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EX Show that

is continuous on .

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EX Show that

is continuous on .

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Rule 1 Assume and are continuous at . Let be a constant. The following functions are cont. at :

and

Remark The rule is true by the limit laws. For example, since are continuous at , it follows that is close to and is close to . So is close to .

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Rule 2 Functions which are continuous at every number in their domains: 1. Polynomials

2. Rational functions

3. Power functions

4. Root functions

If , the function is continuous on . 5. Trig functions

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EX Find the domain for each of the following functions. Explain why they are continuous on their domains?

(1)

(2) (3)

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EX (Use the continuity to find limits) Evaluate

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Def Let and be functions. Define the function by

is called the composite function of and . It is denoted by

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EX with

with

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Limits of composite functions If has the limit as and is a continuous function, then

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EX Evaluate the limit

where

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Continuity of composite functions If is continuous at and is continuous at , then the composite function is continuous at .

Remark This fact follows directly from the definition. In fact, since is cont. at and is cont. at , we get is close to as and is close to

as . So is close

to . Thus is cont. at .

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EX Where are the following functions continuous?

(a)

(b)