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1.2 Finding Limits. Limits described. Goal: To see how a function behaves near a point (such as a discontinuity) use 2 sets of x-values that approach from the left and the right of that point

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limits described
Limits described
  • Goal: To see how a function behaves near a point (such as a discontinuity) use 2 sets of x-values that approach from the left and the right of that point
  • Definition: if f(x) approaches a #, L, as x approaches c from either side, the limit of f(x) as x approaches c is L.
numerical estimation
Numerical estimation
  • Estimate the limit numerically by completing the table for
answer
Answer
  • Limit is L=1 because y-values on both sides of x=2 approach y=1
graphical interpretation
Graphical interpretation
  • Determine the limits at x=1, 2, 3, and 4 given the graph
  • Remember limits occur at and when the y-values are the same as both sides of the c (x-values) approach c; otherwise L=DNE (does not exist)
answer1
Answer
  • The limits for the various c values:
  • C=1: L=1 (both sides approach y=1 despite point at y=2)
  • C=2: L=DNE (left is at y=2 and right is at y=3)
  • C=3: L=1 (both sides approach y=1)
  • C=4: L=2 (both sides approach y=2)
show dne
Show DNE
  • Show limit DNE for
  • Make a table or graph.
answer2
Answer
  • Left L=-1 and right L=1
  • -1≠1 so L=DNE
  • Typical of step functions
show dne1
Show DNE
  • Show limit DNE for
  • Make a table or graph.
answer3
Answer
  • Both sides approach positive infinity which is not a #; therefore, L=DNE
  • Typical of rational functions
show dne2
Show DNE
  • Show limit DNE for
  • Make a table or graph.
answer4
Answer
  • F(x) oscillates between -1 and 1 so L=DNE
  • Typical of odd functions
common behavior associated with dne limits
Common behavior associated with DNE limits
  • F(x) approaches different # from right of c than approaches from left of c.
  • F(x) increases or decreases without bound as x approaches c.
  • F(x) oscillates between 2 fixed values as x approaches c.
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