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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 • 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 • Estimate the limit numerically by completing the table for
Answer • Limit is L=1 because y-values on both sides of x=2 approach y=1
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)
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 limit DNE for • Make a table or graph.
Answer • Left L=-1 and right L=1 • -1≠1 so L=DNE • Typical of step functions
Show DNE • Show limit DNE for • Make a table or graph.
Answer • Both sides approach positive infinity which is not a #; therefore, L=DNE • Typical of rational functions
Show DNE • Show limit DNE for • Make a table or graph.
Answer • F(x) oscillates between -1 and 1 so L=DNE • Typical of odd functions
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.