MATH 131-505 Spring 2015
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2.2
2.2 The Limit of a Function
We write
lim f (x) = L
x→a−
and say the left-hand limit of f (x) as x approaches a (or the limit of f (x) as x approaches
a from the left) is equal to L if we can make the values of f (x) arbitrarily close to L by taking x
to be sufficiently close to a and x < a (or x approaches a from the left hand side).
Similarly, if we require that x > a, we get “the right-hand limit of f (x) as x approaches a is
equal to L” and we write
lim+ f (x) = L
x→a
Example 1: (p. 100) The graph of a function f is shown below. Use it to state the values (if they
exist) of the following:
(a) limx→2− f (x) =
(b) limx→2+ f (x) =
(c) limx→2 f (x) =
(d) limx→5− f (x) =
(e) limx→5+ f (x) =
(f) limx→5 f (x) =
We write
lim f (x) = L
x→a
and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x)
arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side
of a) but not equal to a.
Note: We have “but x 6= a” in the definition of limx→a f (x) = L.
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MATH 131-505 Spring 2015
Theorem:
2.2
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Liu
limx→a f (x) = L ⇐⇒ limx→a− f (x) = limx→a+ f (x) = L
This theorem says that the limit of a function exist if and only if the left hand side limit and the
right hand side limit exist and the limits are equal.
Examples:
2. Consider example 1. Find all a such that limx→a f (x) does not exist.
3. f (x) = x2 − x + 2.
Examples:
4. (p. 96) Guessing the limit of limx→1
x−1
from numerical values.
x2 − 1
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MATH 131-505 Spring 2015
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Liu
5. (p. 103) Sketch the graph of an example of a function f that satisfies all of the given conditions.
lim f (x) = 1,
x→0
lim f (x) = −2,
x→3−
lim f (x) = 2, f (0) = −1, f (3) = 1
x→3+
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