Limits of Functions and Continuity

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a

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x1

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x2

f (a) = L

lim ( )

( )

x af x L

f a exists

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a

f(a) ≠ L

lim ( )

( )

x af x L

f a does not exists

o

The Limit of a FunctionThe limit as x approaches a (x → a) of f (x) = L means that as x gets closer and closer to a (on either side of a), f (x) must approach L. Here, f (a) does not need to exist for the limit to exist.

f (x1)

f (x2)

f (x1)

f (x2)

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x1

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x2

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a

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x

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x

L1

L2

The Limit of a FunctionIf the function values as x approaches a from each side of a do not yield the same function value, the function does not exist.

1

2

lim ( )

lim ( )x a

x a

One Sided Limits

f x L

f x L

Since lim ( ) lim ( ), lim ( ) .x ax a x a

f x f x f x DNE

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a

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x

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x

The Limit of a FunctionIf as x approaches a from either side of a, f (x) goes to either infinity or negative infinity, the limit as x approaches a of f (x) is positive or negative infinity respectively.

lim ( )

lim ( )x a

x a

One Sided Limits

f x

f x

Since lim ( ) lim ( ), then lim ( ) .x ax a x a

f x f x f x

f → ∞

|

x

The Limit of a FunctionIf as x approaches infinity (or negative infinity), f (x) approaches L, then the limit as x approaches a of f (x) is L.

lim ( )x

f x L

L

f (x1)

x → ∞

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a

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b

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a

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b

o

ContinuityA function is continuous over an interval of x values if it has no breaks, gaps, nor vertical asymptotes on that interval.

Continuous on (a, b)

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c

Not Continuous on (a, b) since discontinuous at x = c

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a

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b

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a

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b

o

ContinuityIn other words, a function is continuous at x = c if the following is true.

Continuous on (a, b)

|

c

Not Continuous on (a, b) since discontinuous at x = c

)()(lim cfxfcx

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c

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a

|

b

Continuity Types

Is the following continuous?

Infinite Discontinuity

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c

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a

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b

Jump Discontinuity

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a

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b

o

Continuity TypesIs the function continuous?

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c

Removable Discontinuity

•