2.2 Limits Involving Infinity
Name: ____________________
Objectives: Students will be able to compute limits involving infinity
using various methods.
Let f(x) = 1/x. Find lim f(x) and lim f(x).
x -> ∞
x -> -∞
The line y = b is a ___________ __________ of the graph of a
function y = f(x) if either lim f(x) = b or lim f(x) = b.
x -> ∞
x -> -∞
Apr 15­8:34 PM
Example Use graphs and tables to find lim f(x) , lim f(x), and
x -> ∞
x -> -∞
2
identify all horizontal asymptotes of f(x) = x/√x + 1.
Apr 15­8:52 PM
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The Sandwich Theorem also holds for limits as x -> ±∞.
Example Find lim sinx
x -> ∞
x
Apr 15­9:02 PM
The properties of limits also hold for limits as x -> ±∞.
Example Find lim 5x + sinx .
x
Apr 15­9:03 PM
2
Let's revisit f(x) = 1/x and find lim f(x) and lim f(x).
x -> 0+
x -> 0-
The line x = a is a ________ _________ of the graph of a function
y = f(x) if either lim f(x) = ±∞ or lim f(x) = ±∞.
x -> a+
x -> a-
Example a.) Find the V.A. of the graph of f(x).
b.) Describe the behavior of f(x) to the left and right of each V.A.
f(x) = (x2 - 1)/(2x + 4)
Apr 15­9:12 PM
End Behavior Models For numerically large values of x, we can
sometimes model the behavior of a complicated function by a
simpler one that acts virtually in the same way.
Let f(x) = 3x4 - 2x3 + 3x2 -5x + 6 and g(x) = 3x4. Show that while f
and g are quite different for numerically small values of x, they are
virtually identical for x large.
Apr 16­7:43 AM
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End Behavior Model The function g is:
a.) a right end behavior model for f if and only if lim f(x)/g(x) = 1.
x -> ∞
b.) a left end behavior model for f if and only if lim f(x)/g(x) = 1.
x -> -∞
Examples Find an end behavior model for:
a.) f(x) = 2x5 +x4 - x2 + 1
b.) f(x) = 2x3 - x2 + x - 1
3x2 - 5x + 7
5x3 + x2 + x - 5
Apr 16­7:49 AM
A function's right and left end behavior models need not be the
same function.
Example Find a.) a simple basic function as a right end behavior
model and b.) a simple basic function as a left end behavior model.
f(x) = x2 + e-x
We can investigate the graph of y = f(x) as x -> ±∞ by investigating
the graph of y = f(1/x) as x -> 0.
Example Find lim sin(1/x).
x -> ∞
Homework: Pages 76-77: 1-55 odd, 59-64
Apr 16­7:54 AM
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Apr 18­8:32 AM
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