Introduction to
Limits
OBJECTIVE:
• Calculate a limit
Limits of Function Values
• Definition of Limit:
• If f  x  becomes arbitrarily close to a unique number L as x
approaches c from either side, the limit of f  x  as x
approaches c is L . This is written as lim f  x   L
x c
• Frequently when studying a function, we find ourselves
interested in the function’s behavior near a particular point, but
not at that point.
• Exploring numerically how a function behaves near a particular
point at which we cannot directly evaluate because the function
leads to division by zero.
Finding a limit by using a table of values
• Use a table to estimate numerically the limit.
• 1.
lim  3 x  2 
X
x 2
1.9
1.99
1.999
2
2.001
2.01
2.1
2
2.001
2.01
2.1
f  x
 x2  5x  2
• 2. lim

x 2
X
f  x
1.9
1.99
1.999
Finding a limit by using a table of values
• 3.
x3  x 2  x  1
lim
x 1
x 1
X
0.9
0.99
0.999
1
1.001
1.01
1.1
f  x
• 4. lim
x 0
X
x
x 1 1
-0.01
-0.001
-0.0001
0
0.0001
0.001
f  x
 2 x  1, x  2
lim
• 5. x2 
2 x  2, x  2
X
f  x
1.9
1.99
1.999
2
2.001
2.01
2.1
0.01
Finding a limit by using a graph
• Using a calculator to guess the limit numerically as x gets
closer and closer to c. You discover the behavior of a function
near the x-value at which you are trying to evaluate a limit.
But sometimes calculators can give false values and
misleading impressions for functions that are undefined at a
point or fail to have a limit there, because the calculator
connects pixels and can’t show the infinitely many oscillations
over any interval that contains 0.
x2  1
• 6. lim
x 2 x  1
7. lim
x 0
x
x
Finding a limit by using a graph
• 8.
1
lim 2
x 0 x
• 9.
1
limsin  
x 0
x
• Windows x-min = -0.25 and
x-max = 0.25 and xscl =
0.05, y-min = -1.2 and ymax = 1.2 and yscl = 0.2
Conditions under which Limits
Do Not Exist
• The lim f  x   DNE if any of the following conditions is true.
x c
• It jumps: f  x  approaches a different number from the right
side of c than from the left side of c.
• It grows too “large” or too “small” to have a limit: f  x  increases
or decreases without bound as x approaches c.
• It oscillates to much to have a limit: f  x  oscillates between two
fixed values as x approaches c.
Limit Laws
f  x   L and lim g  x   M
• If L, M , c, and k are real numbers and lim
x c
x c
lim  f  x   g  x    L  M
• 1.
Sum Rule:
• 2.
Difference Rule: lim
 f  x   g  x   L  M
x c
• 3.
 k  f  x   k  L
Constant Multiple Rule: lim
x c
• 4.
Product Rule:
• 5.
Quotient Rule:
• 6.
Power Rule:
• 7.
Root Rule:
x c
lim  f  x   g  x    L  M
x c
f  x
L
lim
 , M 0
x c g  x 
M
n
lim  f  x    Ln , n is a positive integer
x c
1
n
lim  n f  x    L  L , n is a positive integer

x c 
n
EX: Find the following limits
 x  2 x  5
• 10. lim
x 4
x2  2x  5
• 11. lim
x 4
x 1
4 x2  3
• 12. xlim
2
• 13. lim5
x 4
2