2.2 Limits Involving Infinity Finite Limits as – The symbol for infinity does not represent a real...
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Transcript of 2.2 Limits Involving Infinity Finite Limits as – The symbol for infinity does not represent a real...
2.2 Limits Involving Infinity• Finite Limits as– The symbol for infinity does not represent a real
number.– We use infinity to describe the behavior of a function
when the values in its domain or range outgrow all finite bounds.• When we say “the limit of f as x approaches infinity” we mean
the limit of f as x moves increasingly far to the right of the number line.• When we say “the limit of f as x approaches negative infinity”,
we mean the limit of f as x moves increasingly far to the left.
x
• Looking at f(x) = 1/x, we observe:
• We can say that the line y = 0 is a horizontal asymptote of the graph of f.
Horizontal Asymptote
Looking for Horizontal Asymptotes• Use graphs and tables to findand identify all horizontal asymptotes of
• The horizontal asymptotes are y = 1 and y = -1.
lim ( ), lim ( )x xf x f x
2( ) / 1.f x x x
Finding a Limit as x Approaches Infinity
• Find
• The numerator is decreasing to a very small number. The denominator is increasing to a very large number. This causes the function to approach 0.
Using Theorem 5
• Find
Finding Vertical Asymptotes• Find the vertical asymptotes of f(x) = 1 / x².
Describe the behavior to the left and right of each vertical asymptote.
• The values of the function approach infinity on either side of x = 0.
• The line x = 0 is the only vertical asymptote.
Finding Vertical Asymptotes
• The graph of f(x) = tan x = (sin x) / (cos x) has infinitely many vertical asymptotes, one at each point where the cosine is zero.
Modeling Functions for |x| Large• Letand . Show that while f and g are quite
different for numerically small values of x, they are virtually identical for |x| large.
4 3 2( ) 3 2 3 5 6f x x x x x 4( ) 3g x x
Finding End Behavior Models• Find an end behavior model for:a.
b.
Finding End Behavior Models• Let f(x) = x + e-x . Show that g(x) = x is a right end
behavior model for f while h(x) = e-x is a left end behavior model for f.
• On the right,
• On the left,
Using Substitution
• Find limsin(1/ )x
x