Lesson 1-Functions and
Limits
Michael Buenly B. Owogowog
BS Chemical Engineering, Bicol University
Objectives:
• 1. Review functions and their operations;
• 2. Discuss the graphical interpretation of limits;
• 3. Discuss the theorem on limits and apply them to problems.
A. Functions
• When two quantities example x and y are related so
that for some range of values of x, the value of y is
determined by that of x, we say that y is a function of
x.
• 𝑦 = 𝑓(𝑥)
• For a square with side length c, the area is given by:
2
•𝐴 = 𝑐
• Correspondence from a set X of real number 𝑥 to a set 𝑌 of real
number 𝑦 where the number 𝑦 is unique for a specific value of 𝑥
• The set of number 𝑥 is called the domain, while the set of number 𝑦
is called the range
• Can be one-to-one relation or many-to-one relation
A.1.Types of Functions
• Algebraic Functions
• Linear Functions (𝑦 = 𝑚𝑥 + 𝑏)
• Quadratic functions (𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐)
• Polynomial functions (𝑦 = 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−1 + 𝑎2 𝑥 𝑛−2 +
⋯ + 𝑎𝑛−1 𝑥 + 𝑎𝑛 )
• Rational Functions (𝑦 =
𝑃 𝑥
)
𝑄(𝑥)
• Transcendental Functions
• Trigonometric functions (𝑦 = sin 𝑥 , 𝑦 = cos 𝑥 , 𝑦 =
tan 𝑥 , etc)
• Exponential functions (𝑦 = 𝑐𝑎𝑛 , 𝑦 = 𝑐𝑒 𝑛 )
• Logarithmic Functions (𝑦 = 𝑐 log 𝑎 𝑥 )
• Inverse Trigonometric Functions (𝑦 = arcsin 𝑥 , etc)
• Hyperbolic functions (𝑦 = sinh 𝑥 , 𝑒𝑡𝑐. )
1
1
𝑥
−𝑥
• sinh 𝑥 = 𝑒 − 𝑒
; cosh 𝑥 = (𝑒 𝑥 + 𝑒 −𝑥 )
2
2
Special Functions
• Piecewise
• 𝑦 =
𝑓1 𝑥 𝑖𝑓 𝑥 ≥ 𝑎
𝑓2 𝑥 𝑖𝑓 𝑥 < 𝑎
• Absolute value function
• 𝑦 = 𝑥 = 𝑥2
• Greatest integer function
• 𝑦 = 𝑥
• Signum Function
+1 𝑖𝑓 𝑥 > 0
• 𝑦 = sgn 𝑥 = 0 𝑖𝑓 𝑥 = 0
−1 𝑖𝑓 𝑥 < 0
• Inverse Functions(anti-function)
• 𝑦 = ln 𝑥
• Becomes 𝑒 𝑦 = 𝑥
• Applying the inverse
• 𝑒𝑥 = 𝑦
• Even and odd functions
• If even, f(x)=f(-x); odd, f(-x)=-f(x)
• Example of even is 𝑦 = 𝑓(𝑥) = cos 𝑥
• Example of odd is y = f(x) = 𝑥 3
• 𝑓 −2 = −𝑓 2 = −8
A.2.Operations of Functions
• 1. Evaluation
• Given the function 𝑦 = 𝑓 𝑥 = 𝑥 2 + 1, find 𝑓(2)
• 𝑦 = 𝑓 2 = 22 + 1 = 5
1
3
2
• Given the function 𝑦 = 𝑓 𝑥 = 𝑥 + 𝑥 + + 1, find 𝑓(−1)
• 𝑦 = 𝑓 −1 = −1
3
+ −1
2
+
1
(−1)
𝑥
+1=0
• Given the function 𝑦 = 𝑓 𝑥 = cos 2𝑥 − 2 sin 𝑥, find 𝑓(𝜋) and
𝑓 𝑥 − 𝑓(−𝑥)
• Given the function 𝑦 = 𝑓 𝑥 = cos 2𝑥 − 2 sin 𝑥, find 𝑓(𝜋) and
𝑓 𝑥 − 𝑓(−𝑥)
• 𝑦 = 𝑓 𝑥 = cos 2𝑥 − 2 sin 𝑥
• 𝑦 = 𝑓 𝜋 = cos 2𝜋 − 2 sin 𝜋 = 1 − 2 0 = 1
• 𝑓 𝑥 − 𝑓 −𝑥 = cos 2𝑥 − 2 sin 𝑥 − cos −2𝑥 + 2 sin −𝑥
= −4 sin 𝑥
2. Addition and Subtraction: 𝑓 ± 𝑔 𝑥 = 𝑓 𝑥 ± 𝑔(𝑥)
3. Multiplication: 𝑓 • 𝑔 𝑥 = 𝑓 𝑥 • 𝑔(𝑥)
4. Nested function: Circle Operation: 𝑓∘𝑔 𝑥 = 𝑓 𝑔 𝑥
Example 5-10 minutes
• Given 𝑓 𝑥 = 𝑥 and 𝑔 𝑥 = 𝑥 2 + 1, find
• 1. 𝑓 𝑥 + 𝑔 𝑥
• 2. 𝑓 𝑥 • 𝑔(𝑥)
• 3.
𝑔(𝑥)
𝑓(𝑥)
• 4. 𝑓∘𝑔 𝑥
• 5. 𝑓 𝑔 𝑓 𝑥
• 6. 𝑓 𝑓 𝑔 𝑔 𝑥
B.Limits
• Let 𝑓 𝑥 be a function of x and let a be constant. If there is a number
L such that, in order to make the value of 𝑓(𝑥) as close to L as may be
desired, it is sufficient to choose x close enough to a, but different
from a, then we say that the limit of 𝑓 𝑥 , as x approaches a, is L. We
write:
• lim 𝑓(𝑥) = 𝐿
𝑥→𝑎
• The same idea is conveyed when writing:
• As 𝑥 → 𝑎, 𝑓 𝑥 → 𝐿
B.1. Theorems on Limits
• Limit of a constant
• lim 𝑐 = 𝑐
𝑥→𝑎
• lim 𝑐 𝑥 0 = 𝑐 𝑎
0
𝑥→𝑎
=𝑐
• Limit of a function
• lim 𝑓(𝑥) = 𝑓 𝑎 ; 𝑓 𝑎 𝜖𝑅
𝑥→𝑎
• Sum and difference of limits
• If lim 𝑓1 (𝑥) = 𝐿1 , lim 𝑓2 (𝑥) = 𝐿2 … , lim 𝑓𝑛 𝑥 = 𝐿𝑛 , then
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
• lim 𝑓1 (𝑥) ± lim 𝑓2 (𝑥) ± ⋯ lim 𝑓𝑛 𝑥 = 𝐿1 ± 𝐿2 ± ⋯ 𝐿𝑛
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
• Multiplication of limits
• If lim 𝑓1 (𝑥) = 𝐿1 , lim 𝑓2 (𝑥) = 𝐿2 … , lim 𝑓𝑛 𝑥 = 𝐿𝑛 , then
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
• lim 𝑓1 (𝑥) • lim 𝑓2 (𝑥) • ⋯ lim 𝑓𝑛 𝑥 = 𝐿1 • 𝐿2 • ⋯ 𝐿𝑛
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
• Power of a limit
• If lim 𝑓(𝑥) = 𝐿, and n is any positive integer, then
𝑥→𝑎
• lim 𝑓 𝑥
𝑥→𝑎
𝑛
= 𝐿𝑛
• Limits of a sum or difference
• lim 𝑓 𝑥 ± 𝑔(𝑥) = lim 𝑓 𝑥 ± lim 𝑔 𝑥
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
• Limits of a product
• lim 𝑓 𝑥 • 𝑔(𝑥) = lim 𝑓 𝑥 • lim 𝑔 𝑥
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
• Limits of a rational function
• If lim 𝑓(𝑥) = 𝐿, lim 𝑔(𝑥) = 𝑀, then
•
𝑥→𝑎
𝑓(𝑥)
lim
𝑥→𝑎 𝑔(𝑥)
𝑥→𝑎
lim 𝑓(𝑥)
= 𝑥→𝑎
lim 𝑔(𝑥)
=
𝑥→𝑎
𝐿
;𝑔
𝑀
𝑥 ≠0, 𝑀≠0
• Limits of a radical
• If lim 𝑓(𝑥) = 𝐿, and n is any positive integer, then
𝑥→𝑎
• lim
𝑛
𝑥→𝑎
𝑓 𝑥 =
𝑛
𝐿
• Uniqueness Theorem
• If lim 𝑓(𝑥) = 𝐿, and lim 𝑔(𝑥) = 𝑀, then
𝑥→𝑎
• 𝐿=𝑀
𝑥→𝑎
Unit circle
https://www.mathsisfun.com/
algebra/trig-interactive-unitcircle.html
Trigonometric Identities
• sin2 𝑥 + cos2 𝑥 = 1
• sin(𝑎 ± 𝑏) = sin 𝑎 cos 𝑏 ± cos 𝑎 sin 𝑏
• cos(𝑎 ± 𝑏) = cos 𝑎 cos 𝑏 ∓ sin 𝑎 sin 𝑏
• tan(𝑎 + 𝑏) =
tan 𝑎±tan 𝑏
1∓tan 𝑎 tan 𝑏
The number e as a limit
• lim 1 + 𝑛
𝑛→0
1
𝑛
= lim 1 +
𝑛→∞
1 𝑛
𝑛
•𝑒 = 2.718281828459045 …
=𝑒
Special Limits
sin 𝑡
•lim
=1
𝑡→0 𝑡
1−cos 𝑡
•lim
=0
𝑡
𝑡→0
𝑒 𝑡 −1
•lim
=1
𝑡→0 𝑡
Examples finish in 5-10 minutes
1. lim 𝑥 2𝑥 + 1
𝑥→3
𝑥
2. lim
𝑥→4 −7𝑥+1
𝑥 2 −25
3.lim
𝑥→4 𝑥−5
𝑥−4
4.lim
𝑥→4 𝑥−2
2𝑥 2 +𝑥−3
5.lim
𝑥→1 𝑥−1
1−cos 𝑦
6.lim
𝑦→0 sin2 𝑦
sin 𝑦 sin 2𝑦
7. lim
𝑦→0 1−cos 𝑦
3
Study
• One-sided limits
• Infinite limits
• Limits at infinity
• Limit of trigonometric functions