This document defines limits and provides examples of calculating different types of limits. It introduces:
- The definition of a limit as the value a function approaches for a given input value.
- Examples of calculating one-sided (right-hand and left-hand) limits and infinite limits.
- Laws for calculating limits of sums, differences, products, and constant multiples.
- Infinite limits as the function approaches positive even integer powers or from the left/right sides.
2. WHAT IS LIMIT
It is defined as the values that a function approaches the output for the
given input values.
It is defined as the value that the function approaches as it goes to a
variable value.
Let 𝑓 𝑥 be a function defined at all values in an open interval containing
𝑎, with the possible exception of itself, and let 𝐿 be a real number. If all
values of the function 𝑓 𝑥 approaches the real number 𝐿 as the values of
𝑥 ≠ 𝑎 approach the number 𝑎, then we say that the limit of 𝑓 𝑥 as 𝑥
approaches 𝑎 is 𝐿. (In other words, as 𝑥 gets close to 𝑎, 𝑓 𝑥 gets close and
stays close to 𝐿)
lim 𝑓 𝑥 = 𝐿
3. EXAMPLE OF LIMIT
What is the limit of the function 𝑓 𝑥 = 𝑥3 as 𝑥 approaches 3 ?
lim
𝑥→3
𝑓 𝑥 = lim
𝑥→3
𝑥3
Substitute 3 for 𝑥 in the limit function.
lim
𝑥→3
𝑥3 = 33 = 27
4. RIGHT-HAND LIMIT
If 𝑥 approaches 𝑎 from the right side, i.e. from the values greater
than 𝑎, the function is said to have a right-hand limit. If 𝑞 is the
right-hand limit of 𝑓 as 𝑥 approaches 𝑎, we write as
lim
𝑥→𝑎+
𝑓 𝑥 = 𝑞
5. EXAMPLE OF RIGHT-HAND LIMIT
When 𝑥 = 3.1, 𝑓 3.1 = 29.791
When 𝑥 = 3.01, 𝑓 3.01 = 27.270901
When 𝑥 = 3.001, 𝑓 3.001 = 27.027009001
When 𝑥 = 3.0001, 𝑓 3.0001 = 27.002700090001
As 𝑥 decrease and approaches 3, 𝑓 𝑥 still approaches 27.
lim
𝑥→3+
𝑥3 = 27
6. LEFT-HAND LIMIT
If 𝑥 approaches 𝑎 from the left side, i.e. from the values lesser
than 𝑎, the function is said to have a left-hand limit. If 𝑝 is the
right-hand limit of 𝑓 as 𝑥 approaches 𝑎, we write as
lim
𝑥→𝑎−
𝑓 𝑥 = 𝑝
7. EXAMPLE OF LEFT-HAND LIMIT
When 𝑥 = 2.9, 𝑓 2.9 = 24.389
When 𝑥 = 2.99, 𝑓 2.99 = 26.730899.
When 𝑥 = 2.999, 𝑓 2.999 = 26.973008999
When 𝑥 = 2.9999, 𝑓 2.9999 = 26.997300089999
As 𝑥 increase and approaches 3, 𝑓 𝑥 still approaches 27.
lim
𝑥→3−
𝑥3 = 27
8. BASIC RULE FOR LIMIT
For any real number 𝑎 and any constant 𝑐,
lim
𝑥→𝑎
𝑥 = 𝑎
lim
𝑥→𝑎
𝑐 = 𝑎
For example:
1)
lim
𝑥→2
𝑥
Substitute 2 for 𝑥 in the limit
function.
lim
𝑥→2
𝑥 = 2
2)
lim
𝑥→2
5
The limit of a constant is that
constant.
lim
𝑥→2
5 = 5
9. SUM LAW FOR LIMIT
Let 𝑓 𝑥 and 𝑔 𝑥 be defined for all 𝑥 ≠ 𝑎 over some open interval containing 𝑎. Assume that
𝐿 and 𝑀 are real numbers such that lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and lim
𝑥→𝑎
𝑔 𝑥 = 𝑀. Let 𝑐 be a constant.
lim
𝑥→𝑎
𝑓 𝑥 + 𝑔 𝑥 = lim
𝑥→𝑎
𝑓 𝑥 + lim
𝑥→𝑎
𝑔 𝑥 = 𝐿 + 𝑀
For example:
Evaluate lim
𝑥→−3
𝑥 + 3
Use the sum law for limit, lim
𝑥→𝑎
𝑓 𝑥 + 𝑔 𝑥 = lim
𝑥→𝑎
𝑓 𝑥 + lim
𝑥→𝑎
𝑔 𝑥
lim
𝑥→−3
𝑥 + 3 = lim
𝑥→−3
𝑥+ lim
𝑥→−3
3
Use the basic rule for limit, lim
𝑥→𝑎
𝑥 = 𝑎 and lim
𝑥→𝑎
𝑐 = 𝑎
lim
𝑥→−3
𝑥+ lim
𝑥→−3
3 = −3 + 3 = 0
10. DIFFERENCE LAW FOR LIMIT
Let 𝑓 𝑥 and 𝑔 𝑥 be defined for all 𝑥 ≠ 𝑎 over some open interval containing 𝑎. Assume that 𝐿 and 𝑀 are
real numbers such that lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and lim
𝑥→𝑎
𝑔 𝑥 = 𝑀. Let 𝑐 be a constant.
lim
𝑥→𝑎
𝑓 𝑥 − 𝑔 𝑥 = lim
𝑥→𝑎
𝑓 𝑥 − lim
𝑥→𝑎
𝑔 𝑥 = 𝐿 − 𝑀
For example:
Evaluate lim
𝑥→3
𝑥 − 5
Use the difference law for limit, lim
𝑥→𝑎
𝑓 𝑥 − 𝑔 𝑥 = lim
𝑥→𝑎
𝑓 𝑥 − lim
𝑥→𝑎
𝑔 𝑥
lim
𝑥→3
𝑥 − 3 = lim
𝑥→3
𝑥 − lim
𝑥→3
5
Use the basic rule for limit, lim
𝑥→𝑎
𝑥 = 𝑎 and lim
𝑥→𝑎
𝑐 = 𝑎
lim
𝑥→3
𝑥 − lim
𝑥→3
5 = 3 − 5 = −2
11. PRODUCT LAW FOR LIMIT
Let 𝑓 𝑥 and 𝑔 𝑥 be defined for all 𝑥 ≠ 𝑎 over some open interval containing 𝑎. Assume that
𝐿 and 𝑀 are real numbers such that lim
𝑥→𝑎
𝑓 𝑥 = 𝐿 and lim
𝑥→𝑎
𝑔 𝑥 = 𝑀. Let 𝑐 be a constant.
lim
𝑥→𝑎
𝑓 𝑥 ∙ 𝑔 𝑥 = lim
𝑥→𝑎
𝑓 𝑥 ∙ lim
𝑥→𝑎
𝑔 𝑥 = 𝐿 ∙ 𝑀
For example:
Evaluate lim
𝑥→3
𝑥 𝑥 + 5
Use the product law for limit, lim
𝑥→𝑎
𝑓 𝑥 ∙ 𝑔 𝑥 = lim
𝑥→𝑎
𝑓 𝑥 ∙ lim
𝑥→𝑎
𝑔 𝑥
lim
𝑥→3
𝑥 𝑥 + 5 = lim
𝑥→3
𝑥 ∙ lim
𝑥→3
𝑥 + 5
Use the sum law for limit, lim
𝑥→𝑎
𝑝 𝑥 + 𝑞 𝑥 = lim
𝑥→𝑎
𝑝 𝑥 + lim
𝑥→𝑎
𝑞 𝑥
lim
𝑥→3
𝑥 ∙ lim
𝑥→3
𝑥 + 5 = lim
𝑥→3
𝑥 ∙ lim
𝑥→3
𝑥 + lim
𝑥→3
5
Use the basic rule for limit, lim
𝑥→𝑎
𝑥 = 𝑎 and lim
𝑥→𝑎
𝑐 = 𝑎
lim
𝑥→3
𝑥 ∙ lim
𝑥→3
𝑥 + lim
𝑥→3
5 = 3 3 + 5 = 3 8 = 24
12. CONSTANT MULTIPLE LAW FOR LIMIT
Let 𝑓 𝑥 be defined for all 𝑥 ≠ 𝑎 over some open interval containing 𝑎. Assume that 𝐿 are real numbers
such that lim
𝑥→𝑎
𝑓 𝑥 = 𝐿. Let 𝑐 be a constant.
lim
𝑥→𝑎
𝑐𝑓 𝑥 = 𝑐 lim
𝑥→𝑎
𝑓 𝑥 = 𝑐𝐿
For example:
Evaluate lim
𝑥→3
5𝑥2
.
Use the constant multiple law for limit, lim
𝑥→𝑎
𝑐𝑓 𝑥 = 𝑐 lim
𝑥→𝑎
𝑓 𝑥
lim
𝑥→3
5𝑥2
= 5 lim
𝑥→3
𝑥2
Substitute 5 for 𝑥 in 5 lim
𝑥→3
𝑥2
.
5 lim
𝑥→3
𝑥2
= 5 32
= 5 9 = 45
Type equation here.
13. INFINITE LIMIT FROM THE LEFT
Let 𝑓 𝑥 be a function defined at all values in an open interval of the form
𝑏, 𝑎 .
If the values of 𝑓 𝑥 increase without bound as the values of 𝑥 (where 𝑥 < 𝑎)
approach the number 𝑎, then we sat that the limit as 𝑥 approaches 𝑎 from the
left is positive infinity.
lim
𝑥→𝑎−
𝑓 𝑥 = −∞
If the values of 𝑓 𝑥 decrease without bound as the values of 𝑥 (where 𝑥 < 𝑎)
approach the number 𝑎, then we sat that the limit as 𝑥 approaches 𝑎 from the
left is negative infinity.
lim
𝑥→𝑎−
𝑓 𝑥 = +∞
14. EXAMPLE OF CASE 1
Evaluate the limit lim
𝑥→0−
−
1
𝑥
, if possible.
Here, 𝑓 𝑥 = −
1
𝑥
When 𝑥 = −0.1, 𝑓 0.1 = 10
When 𝑥 = −0.01, 𝑓 0.01 = 100
When 𝑥 = −0.001, 𝑓 0.001 = 1000
When 𝑥 = −0.0001, 𝑓 0.0001 = 10,000
The value of 𝑓 𝑥 increase without bound as 𝑥 approaches 0 from the left.
lim
𝑥→0−
−
1
𝑥
= +∞
15. EXAMPLE OF CASE 2
Evaluate the limit lim
𝑥→0−
1
𝑥
, if possible.
Here, 𝑓 𝑥 =
1
𝑥
When 𝑥 = −0.1, 𝑓 0.1 = −10
When 𝑥 = −0.01, 𝑓 0.01 = −100
When 𝑥 = −0.001, 𝑓 0.001 = −1000
When 𝑥 = −0.0001, 𝑓 0.0001 = −10,000
The value of 𝑓 𝑥 decrease without bound as 𝑥 approaches 0 from the left.
lim
𝑥→0−
1
𝑥
= −∞
16. INFINITE LIMIT FROM THE RIGHT
Let 𝑓 𝑥 be a function defined at all values in an open interval of the form
𝑎, 𝑐 .
If the values of 𝑓 𝑥 increase without bound as the values of 𝑥 (where 𝑥 > 𝑎)
approach the number 𝑎, then we sat that the limit as 𝑥 approaches 𝑎 from the
right is positive infinity.
lim
𝑥→𝑎+
𝑓 𝑥 = +∞
If the values of 𝑓 𝑥 decrease without bound as the values of 𝑥 (where 𝑥 > 𝑎)
approach the number 𝑎, then we sat that the limit as 𝑥 approaches 𝑎 from the
right is negative infinity.
lim
𝑥→𝑎+
𝑓 𝑥 = −∞
17. EXAMPLE OF CASE 1
Evaluate the limit lim
𝑥→0+
1
𝑥
, if possible.
Here, 𝑓 𝑥 =
1
𝑥
When 𝑥 = 0.1, 𝑓 0.1 = 10
When 𝑥 = 0.01, 𝑓 0.01 = 100
When 𝑥 = 0.001, 𝑓 0.001 = 1000
When 𝑥 = 0.0001, 𝑓 0.0001 = 10,000
The value of 𝑓 𝑥 increase without bound as 𝑥 approaches 0 from the right.
lim
𝑥→0+
1
𝑥
= +∞
18. EXAMPLE FOR CASE 2
Evaluate the limit lim
𝑥→0+
−
1
𝑥
, if possible.
Here, 𝑓 𝑥 = −
1
𝑥
When 𝑥 = 0.1, 𝑓 0.1 = −10
When 𝑥 = 0.01, 𝑓 0.01 = −100
When 𝑥 = 0.001, 𝑓 0.001 = −1000
When 𝑥 = 0.0001, 𝑓 0.0001 = −10,000
The value of 𝑓 𝑥 decrease without bound as 𝑥 approaches 0 from the right.
lim
𝑥→0+
−
1
𝑥
= −∞
19. INFINITE LIMITS FROM POSITIVE EVEN
INTEGERS
If 𝑛 is a positive even integer, then
lim
𝑥→𝑎
1
𝑥 − 𝑎 𝑛
= +∞
lim
𝑥→𝑎+
1
𝑥 − 𝑎 𝑛
= +∞
lim
𝑥→𝑎−
1
𝑥 − 𝑎 𝑛
= +∞
23. LIMIT AT INFINITY FOR RATIONAL
FUNCTION
For rational function 𝑓 𝑥 =
𝑝 𝑥
𝑞 𝑥
, the limit at infinity is determined by
the relationship between the degree of 𝑝 and 𝑞.
If the degree of 𝑝 is less than the degree of 𝑞, then the line 𝑦 = 0 is a
horizontal asymptote for 𝑓.
If the degree of 𝑝 is equal to the degree of 𝑞, then the line 𝑦 =
𝑎𝑛
𝑏𝑛
is a
horizontal asymptote for 𝑓, where 𝑎𝑛 and 𝑏𝑛 are the leading
coefficients of 𝑝 and 𝑞.
If the degree of 𝑝 is greater than the degree of 𝑞, then 𝑓 approaches ∞
or −∞ at each end.