Section 1.3
The Limit of a Function
Motivation
Suppose we are Galileo, and we are dropping cannon balls off the
Tower of Pisa. So, we drop the balls and note how far they fall after
various amounts of time.
Time (in seconds) Length (in meters)
1s
4.9 m
2s
19.6 m
3s
44.1 m
4s
78.4 m
5s
122.5 m
6s
176.4 m
Question:
How fast is the ball falling
after exactly 5 seconds?
Motivation Continued
We can easily calculate the average velocity over a duration.
Time (in seconds) Length (in meters)
1s
4.9 m
2s
19.6 m
3s
44.1 m
4s
78.4 m
5s
122.5 m
6s
176.4 m
average velocity =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
𝑡𝑖𝑚𝑒 𝑒𝑙𝑎𝑝𝑠𝑒𝑑
Motivation Continued
Idea: Maybe we can find the real velocity at 5 seconds by letting the
duration get smaller and smaller.
Duration
Average Velocity
5≤𝑡≤6
53.9 m/s
5 ≤ 𝑡 ≤ 5.1
49.49 m/s
5 ≤ 𝑡 ≤ 5.01
49.049 m/s
5 ≤ 𝑡 ≤ 5.001
49.0049 m/s
It looks like the velocity is getting closer and closer to 49 as the
duration gets smaller. Let’s be more precise.
Conceptual Definition of Limits
Suppose 𝑓 𝑥 is defined when 𝑥 is near the number 𝑎 (not
necessarily at 𝑎). Then, we write
lim 𝑓 𝑥 = 𝐿
𝑥→𝑎
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.
Example
𝑥2 − 1
lim
𝑥→1 𝑥 − 1
Find the limit using a graph.
Example
Let 𝑓 be a function defined below. Find lim 𝑓 𝑥 .
𝑥→0
𝑥,
𝑥≠0
𝑓 𝑥 =ቊ
1,
𝑥=0
Example: One-sided limits
−1, 𝑥 < 0
𝑓 𝑥 =ቊ
1, 𝑥 ≥ 0
What is lim 𝑓(𝑥)?
𝑥→0
One-sided Limits Notation
The limit of 𝑓 𝑥 as 𝑥 approaches 𝑎 from the right:
lim+𝑓 𝑥
𝑥→𝑎
The limit of 𝑓 𝑥 as 𝑥 approaches 𝑎 from the left:
lim−𝑓 𝑥
𝑥→𝑎
So, lim 𝑓 𝑥 = 𝐿 if and only if lim+ 𝑓 𝑥 = 𝐿 and lim− 𝑓 𝑥 = 𝐿.
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
Caveats: Example
Let 𝑓 𝑡 =
𝑡 2 +9−3
.
𝑡2
Find lim 𝑓(𝑡). Guess the limit using approximations.
𝑡→0
Let’s try to make a table.
𝒕
𝒇(𝒕)
±1
0.16228
±0.1
0.16662
±0.01
0.16667
Caveats: Example 1 Continued
Let’s keep filling in our table with a calculator.
𝒕
𝒇(𝒕)
±1
0.16228
±0.1
0.16662
±0.01
0.16667
⋮
⋮
±0.00001
0.16700
±0.000001
0.20000
±0.0000001
0.00000
Caveats: Example 2
Find lim sin
𝑥→0
𝒙
1
𝝅
𝒔𝒊𝒏
𝒙
0
1/2
0
1/3
0
1/4
0
⋮
⋮
1/100
0
⋮
⋮
𝜋
𝑥
.
Caveats: Example 2 Continued
Find lim sin
𝑥→0
𝒙
1
𝜋
𝑥
.
𝝅
𝒔𝒊𝒏
𝒙
0
1/2
0
1/3
0
1/4
𝒙
2
𝝅
𝒔𝒊𝒏
𝒙
1
2/3
-1
2/5
1
0
2/7
-1
⋮
⋮
⋮
⋮
1/100
0
2/101
1
⋮
⋮
⋮
⋮
But…
Caveats: Example 2 Continued
𝜋
𝑦 = sin
𝑥
Oscillating behavior
More Precise Definition of Limits
Let 𝑓 be a function defined on some open interval that contains the number 𝑎, except
possibly at 𝑎 itself. Then, we say the limit of 𝑓(𝑥) as 𝑥 approaches 𝑎 is 𝐿, and we write
lim 𝑓 𝑥 = 𝐿
𝑥→𝑎
if for every number 𝜀 > 0 there is a corresponding number 𝛿 > 0 such that
𝑓 𝑥 −𝐿 <𝜀
whenever
0 < 𝑥 − 𝑎 < 𝛿.
Example
Prove lim 2𝑥 + 3 = 9
𝑥→3
Example
Let 𝑓 𝑥 = 𝑥. Find the largest number 𝛿 such that if
𝑥 − 4 < 𝛿, then 𝑥 − 2 < 0.4.
Extra Problems
Try to do these on your own for extra practice. The solutions are given in the
completed notes.
Exercise
Guess the value of the limit (if it exists) by evaluating the function at
the given numbers.
sin 𝑥
lim
𝑥→0 𝑥 + tan 𝑥
𝑥 = ±1, ±0.1, ±0.01
Exercise
Use the given graph of 𝑓 𝑥 = 𝑥 2 to find the largest number 𝛿 such
that
2
if 𝑥 − 1 < 𝛿 then 𝑥 − 1 <
1
.
2