1.2: Introduction to Limits – Graphically & Numerically
“I can…
Use the definition of limit to make an estimate
Determine if a limit of a function does not exist.”
I. Limit
A. Definition: 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.
B. Write the limit like:
II. Techniques to Evaluate Limits
A. Table (Numerical approach) – Sometimes it is difficult to find a limit by
algebraic approaches. Plug the function into ____. Go to __ ____ # __ VALUE. Plug in
the c value (the number __ approaches) and 3 numbers slightly _________ and 3 slightly
________. It does not matter what the limit at c is but it does matter what is to the
immediate left and right of that number. What is a close estimate as we move to the
middle?
B. Graph (Graphical approach)– When looking on a graph, it does ____ matter
what the function is at __ (as long as it exists) but you must look to the _____ and
_______ of c and these values must match.
C. Direct Substitution (Analytic approach)– Sometimes you can substitute c in
without causing division by __ or __________ inside a _________ root.
NOTE: When Trig. functions are used, put your calculator in _________ mode!!!
D. Limits That Fail to Exist
1. Left and Right Differs – In a step function the left and right sides ___ _____ match.
2. Unbounded – There is a vertical ____________ at c and both sides approach __.
3. Oscillating Behavior – A function like ______ sin(1/x) oscillates between 1 and -1.
1. lim x
x 0
2. lim
x 0
1
x2
1

x
3. lim sin 
x 0
III. Model Problems
Guided Practice
Complete the table for lim
x 0
x
-.01
-.001
-.0001
On Your Own
3x  2
x2 x  2
x
x 1 1
.0001
.001
Complete the table for lim
.01
x
1.9
1.99
1.999
2.001
f(x)
f(x)
Find the limit lim x2
Find the limit lim 5 x
Find the limit lim  x cos x 
Find the limit lim
x4
x 4
x 
Find the limit lim
x 0
x 
x
x
(You may wish to graph
it)
 1 
Find the limit lim  2 
x 0 x
 
tan x
x
2, x  3
f ( x)  
Find the limit
4, x  3
lim f ( x)
x 3
x2  x  6
x 1
x3
Find the limit lim
2.01
2.1
Academic Practice
AP Calculus AB
1 – 7: Complete the table and use the result to estimate the limit. Use a graphing calculator to estimate
your results.
1. lim
x 4
x
x4
x  3x  4
3.9
3.99
3.999
x6  6
x
3. lim
2
x 0
4.001
4.01
x
4.1
-0.1
-0.01
-0.001
0.001
0.01
0.1
-0.1
-0.01
-0.001
0.001
0.01
0.1
f(x)
f(x)
1
1

5. lim x  1 4
x 3
x 3
x
2.9
2.99
7. lim
x 0
2.999
3.001
3.01
sin x
x
x
3.1
f(x)
f(x)
9 – 13: Create a table of values for the function and use the result to estimate the limit using a graphing
calculator.
9. lim
x 1
x2
2
x  x6
11. lim
x 1
x4  1
x6  1
sin  2 x 
x 0
x
13. lim
15 – 25: Find the limit (if it exists). If the limit does not exist, explain why.
lim f ( x)
15. lim  4  x 
x 3
21. limsin  x 
x 1
x 2
17.
 4  x, x  2
f ( x)  
0, x  2
23. lim cos
x 0
1
x
19. lim
x2
x2
x2
 x 2  3, x  1

25. 2, x  1

 x, x  1
(a) f(1) =
(b) lim f ( x )
x 1
* Larson and Edwards: Calculus of a Single Variable 9th ed, Brooks/Cole Cengage Learning, 2015, pg 55