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Continuity of Trig and InverseFunctions

Objective: To use limits to define

continuity in trig/inverse functions

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Theorem 1.6.1

If c is any number in the natural domain ofthe stated trigonometric function, then

cxcx

sinsinlim =

cxcx

tantanlim =

cxcx

coscoslim =

cxcx

cotcotlim =

cxcx

secseclim =

cxcx

csccsclim =

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!am"le 1

Find the limit

1

1coslim2

1 x

xx

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!am"le 1

Find the limit

The limit of the cosine is the cosine of thelimit.

1

1coslim2

1 x

xx

( ) 2cos)1(limcos1

1limcos

1

1coslim

1

2

1

2

1

=+=

=

xx

x

x

x

xxx

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Theorem 1.6.#

If f is a one-to-one function that iscontinuous at each point of its domain,then f -1 is continuous at each point of its

domain, that is f -1is continuous at eachpoint of the range of f.

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!am"le $

Where is the function

continuous? 4

lntan)(

2

1

+=

x

xxxf

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!am"le $

Where is the function

continuous?

A fraction is continuous where thenumerator and denominator arecontinuous and the denominator is not

ero.

4

lntan)(

2

1

+=

x

xxxf

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!am"le $

Where is the function

continuous?

A fraction is continuous where thenumerator and denominator are continuousand the denominator is not ero.

The numerator is continuous for ! " #$why?% and the denominator is continuouse&erywhere

4

lntan)(

2

1

+=

x

xxxf

2x

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!am"le $

Where is the function

continuous?

A fraction is continuous where the numeratorand denominator are continuous and thedenominator is not ero.

The numerator is continuous for ! " # $why?%and the denominator is continuouse&erywhere

The function f is continuous for ! " #, not '.

4

lntan)(

2

1

+=

x

xxxf

2x

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The %&uee'ing Theorem

(et f, g, and h be functions satisfying

for all ! in some open inter&al containing the

number c, with the possible e!ception thatthe ine)ualities need not hold at c. If g and hha&e the same limit as ! approaches c, say

then f also has this limit

as ! approaches c, that is

)()()( xhxfxg

Lxhxg cxcx == )(lim)(lim

Lxfcx

=

)(lim

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Theorem 1.6.(

1sin

lim)(0

= x

xa

x0

cos1lim)(

0=

x

xb

x

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!am"le )

Find*x

xx

tanlim0

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!am"le )

Find*x

xx

tanlim0

111cos

1lim

sinlim

cos

1sinlim

000==

=

xx

x

xx

x

xxx

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!am"le )

Find*x

xx

2sinlim

0

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!am"le )

Find*x

xx

2sinlim

0

2122

2sinlim2

2

2sin2lim

00===

x

x

x

x

xx

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!am"le )

Find*x

x

x 5sin

3sinlim

0

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!am"le )

Find*x

x

x 5sin

3sinlim

0

x

xx

x

x 5sin

3sin

lim0

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!am"le )

Find*x

xx 5sin

3sinlim

0

x

xx

x

x

xx

x

xx

5

5sin5

3

3sin3

lim5sin

3sin

lim00

=

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!am"le )

Find*x

xx 5sin

3sinlim

0

x

xx

x

x

xx

x

x

xx

x

xxx

5

5sin3

3sin

lim5

3

5

5sin5

3

3sin3

lim5sin

3sin

lim000

==

5

3

1

1

5

3=

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*ome+or,

+ection 1.

ages1'-1'

1-/ odd