FINDING LIMITS GRAPHICALLY AND NUMERICALLY

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FINDING LIMITS GRAPHICALLY AND NUMERICALLY

• Section 1.2

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When you are done with your homework, you should
be able to

• Estimate a limit using a numerical or graphic
  approach
• Learn different ways that a limit can fail to
  exist
• Study and use a formal definition of limit

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AN INTRODUCTION TO LIMITS

• Consider the function
  

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What is the domain?

• Both C and D

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Consider the limit of this function as x
approaches 2.

• When graphing the function, we could factor the
  numerator and cancel, keeping in mind there will
  be a break in the graph at .
• So we have
• which is the graph of a line with an open circle
  at .
• Lets first estimate the limit as x approaches 2
  graphically. Or, writing this mathematically,

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Evaluate the function at x 2

• 1
• undefined

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Evaluate the limit of the function as x
approaches 2 graphically.

• 1
• DNE

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Now, lets estimate
numerically using a table of values.

• Since is approaching 1 from the left and
  right of 2, we may conclude that

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LIMITS THAT FAIL TO EXIST

• Behavior that differs from the right to the left.
• Consider

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What is the limit of the function as x approaches
0 from the right?

• 0
• Does not exist
• 2

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What is the limit of the function as x approaches
0 from the left?

• 0
• Does not exist
• 2

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What is the limit of the function as x approaches
0?

• 0
• Does not exist
• 2

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UNBOUNDED BEHAVIOR

• Consider the function This is a
  hyperbola with a vertical asymptote at
• Notice that approaching x from either the left or
  the right of 0, f increases without bound, that
  is, f is approaching infinity, which is not an
  actual number.
• Therefore, we say the finite limit does not
  exist.

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OSCILLATING BEHAVIOR

• Consider the function
• Lets examine what happens as x approaches 0.

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Common Types of Behavior Associated with
Nonexistence of a Limit

• approaches a different number from the
  right side of c than it approaches from the left
  side.
• increases or decreases without bound as
  x approaches c.
• oscillates between two fixed values as
  x approaches c.

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Formal Definition of Limit

• Epsilon and delta
• Let f be a function defined on an open interval
  containing c (except possibly at c) and let L be
  a real number. The statement
  means that for each there exists a
  such that if then

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Finding a for a given

• Given the limit
• find delta given that epsilon is 0.01.
• So where should you start?
• Hint Math is all about definitions!