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c Roberto Barrera, Fall 2015
Math 142 3.1 Limits
Limit of a Function
Limit: Suppose the function f (x) is defined for all values of x near a, but not necessarily
at a. If as x approaches a, f (x) approaches the number L, then we say that L is the limit
of f (x) as x approaches a and write
lim f (x) = L.
x→a
lim f (x):
x→a−
lim f (x):
x→a+
Existence of a Limit:
c Roberto Barrera, Fall 2015
Math 142 2
Estimating limits on a calculator
1. Press Y = and enter in the function in Y1 .
2. Graph the function by pressing GRAPH .
3. To check values near a point, press 2nd T RACE (calc) and choose VALUE by
pressing 1 .
4. Enter in desired number to evalate at.
Note: if your calculator is giving the result as a fraction, press MODE , scroll down to
ANSWERS: and select DEC.
Example: Find limx→1− f (x) and limx→1+ f (x) where

x<1
 x,
x=1 .
f (x) = 2,

x − 1,
x>1
Example: Find lim 3x + 1 it it exists.
x→2
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c Roberto Barrera, Fall 2015
Math 142 2
−5x−2
Example: Find lim 3x x−2
it it exists.
x→2
1
it it exists.
Example: Find lim− 1−x
x→1
Example: Find lim f (x) it it exists where
x→0
f (x) =
−x
x+1
x<0
x>0
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c Roberto Barrera, Fall 2015
Math 142 Rules for Limits
The Limit of a Polynomial Function: If p(x) is any polynomial and a is any number,
then
lim p(x) = p(a).
x→a
Rules for Limits: Assume that
lim f (x) = L
x→a
1. lim c f (x) =
x→a
2. lim ( f (x) ± g(x)) =
x→a
3. lim ( f (x) · g(x)) =
x→a
f (x)
4. lim g(x) =
x→a
5. lim ( f (x))n =
x→a
2
Example: Evaluate lim xx2+11
.
−4
x→3
and
lim g(x) = M.
x→
c Roberto Barrera, Fall 2015
Math 142 Limits and Continuity
Continuity:
Definition of continuity implies that a function is continuous at x = a only if
1.
2.
3.
Polynomials:
Rational functions
Example: Determine all points at which the following function is continuous
(
x5 − x4 + x3 + 2x − 1,
x≤1
f (x) = x2 +x+2
.
,
x
>
1
x+1
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