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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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