LIMITS

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LIMITS

• Chris Floyd
• Zach Hilley
• Aaron Hevenstone

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LIMITS

• A LIMIT IS THE VALUE A FUNCTION APPROACHES AT A
  CERTAIN POINT.

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Is it continuous?
Functions can have continuities, discontinuities,
or even bothinuities. Testing for continuities
within a function is similar to testing if a
limit even exists! There are three different
stipulations (big word!) that MUST be met in
order for a function to be continuous.
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Continuity

• Continuous graphs look very smooth and connected.
  They have no holes, breaks, jumps, etc.
• Which graph is continuous?

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So What does it really mean?

• A function f(x) is continuous at xc if THREE
  conditions are met
• Lim f(x) exists
• f(c) is defined
• Lim f(x)f(c)

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UH OH! It didnt pass one of the three tests

• When a limit doesnt meet one of the conditions,
  it has a discontinuity.
• There are THREE main discontinuities
• Jump usually a piecewise function whose pieces
  dont meet up.
• Point When the function has a hole. (a point
  is missing!!!)
• Infinite When the function has neither a limit
  nor is defined at the given x-value. This is
  usually a vertical asymptote.

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Removable vs. Nonremovable

• A discontinuity can be labeled as removable when
  it is a hole in the function. In other words,
  it is when the limit is defined but the value
  f(c) is not. (
• A nonremovable continuity exists when the limit
  does not.

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PRACTICE (CONTINUITY)
Determine whether of not g(x) is continuous at
x1 3x2-x-2 , x?1 g(x) x-1
-2, x1
Give all x-values for which the function is
discontinuous, and classify each instance of
discontinuity. g(x)2x25x-25 x2-25
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Evaluating Limits
The easiest way to evaluate a limit is simply to
plug in the approached value to the
function. lim x2/(x2)??? x -gt 4
If you said 8/3 you are a winner! There are a
few situations where this method falls short. A
limit may not exist, it may have an indeterminant
form, or it may not be equal to the function.
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HOW can a limit not equal the function!??
Because a limit only means the value that a
function approaches, the limit may be equal to a
value different from that of the function, even
if the function does not exist at that location.
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One sided limits
Limits can sometimes be taken from only one side
of a point. If for example you are taking a
limit from the right, you look at what the
function approaches from the right side,
sometimes limits from the right and left are
different. In this case the limit does not exist.
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When does a limit not exist?
A limit does not exist if The limit from the
right does not equal the limit from the
left. The limit approaches infinity. The
function oscillates.
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Approaching infinity
If you are taking a limit of a function as it
approaches infinity look at the degree of the
numerator and denominator. If they are the same,
then the limit is the ratio of the
coefficients. If the degree of the numerator is
greater, the function approaches infinity. If
the degree of the denominator is greater, the
function approaches 0.
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Indeterminant forms

• If a limit approaches certain values that cannot
  be evaluated (but are NOT undefined!) like
  0infinity, 0/0, infinity/infinity, etc. then
  more work must be done to evaluate the limit.
  1/0 IS NOT INDETERMINANT!

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LHopitals Rule (or Little Hospitals) LHopital
s Rule is a useful method for evaluating an
indeterminant limit. To use Little Hospitals
take the derivative of the numerator and the
denominator separately (do not use division rule)
and then evaluate the limit. This method will
NOT work unless the limit is indeterminant.
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Practice!
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Intermediate Value Theorem

• If a function f is continuous on a closed
  interval a,b, and c is any number between f(a)
  and f(b) inclusive, then there is at least one
  number x in the closed interval such the f(x)c.

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• Basically, if f(x) is continuous from on point,
  a, to another, b, then there is a point in
  between them, c.

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Corollary

• f(x) is continuous on a,b and if f(a) and f(b)
  are nonzero numbers and have opposite signs, f(x)
  has a zero in (a,b)
• Theres a zero if it is continuous and goes from
  negative to positive.

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Problems!
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Asymptotes of Limits

• The behavior of a function f(x) as either x or
  f(x) approaches positive or negative infinity.
• There exists a vertical asymptote if the limit,
  as x approaches L from the right or left, of f(x)
  is positive or negative infinity.
• There exists a horizontal asymptote if the limit,
  as x approaches positive or negative infinity,
  of f(x) equals L.

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Fun Stuff

• If the exponents of the variables in the
  numerator and the denominator are equal, then the
  limit approaches the coefficients of the
  variables.
• If the exponent of the variable in the
  denominator is greater, then the limit approaches
  pos. or neg. infinity
• If the exponent of the variable in the numerator
  is greater, then the limit approaches infinity.

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More Problems!
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Limit Definition of the dewhat?

• is the slope of a tangent
  line!Lim f(xh) f(x)
• h-gt0 h

Derivative
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Ive got problems (Limit Definition)
Find the derivative of f(x)5x27x -6 and use it
to calculate f(-1)
Find the derivative of g(x)x2-3x 4 and then
evaluate f(2)