BA

CLS
B A S C A L
 LIMIT OF A FUNCTION
BASIC CALCULUS : WEEK 1

Methods of Existence of limit

Finding Limit of a of a function
Function Limit exists if and only if the
limits from the left and from
the right side exist and are
TABLE OF VALUES equal

→
The 𝑙𝑖𝑚 𝑥 𝑐 𝑓(𝑥) does not exist
(DNE) whenever:

The one-sided limits are not

equal
we only need to go very close to the constant
but will not necessarily go to the constant The function does not
itself. approach a finite value.
The function does not
approach a particular value
GRAPH (oscillation)

The exclusion of a value from

the domain of a function does
not prohibit the evaluation of
the limit of that function at
that excluded value, provided
that 𝑓 is defined at the points
near 𝑐.
In short, pag DNE ang f(c),
→
does not mean na does not
exist din yung 𝑙𝑖𝑚 𝑥 𝑐 𝑓(𝑥)

Observe the 2 sided limits of the graph then

check if they're approaching the same value,
if not then Limit DNE
If there is a hole in the graph at the value
that x is approaching, with no other point for
a different value of the function, then the
limit does still exist - END -
go madumb lesson 1 ka
(kung kelan tapos an quiz, tska ko lang na palang pero parang wala
gets to TT welp) kanang energy
 BASIC LIMIT
LAWS
BASIC CALCULUS : WEEK 2

SUM OR DIFFERENCE LAW

The limit of the product of two
Let 𝑐, 𝑘, 𝑛, 𝐿, and 𝑀 be real numbers, functions is equal to the
and let 𝑓 and 𝑔 be functions such that product of the limits of the
two functions.
𝑙𝑖𝑚 𝑥→𝑐𝑓(𝑥) = 𝐿 and 𝑙𝑖𝑚 𝑥→𝑔𝑐 (𝑥)= 𝑀

If 𝑓(𝑥) is a polynomial, then

QUOTIENT LAW
The limit of the quotient of two
functions is equal to the quotient
LIMIT OF A CONSTANT of the limits of the two functions,
FUNCTION bawal maging zero yung
denominator/devisor
The limit of a constant
function is the constant itself.
Let 𝑘 be any constant,

POWER LAW
The limit of the integral power of a
LIMIT OF AN IDENTITY function is equal to the integral
FUNCTION provided that 𝑛 ∈
power of the limit of the function,
ℤ+
The limit of the identity
function 𝑓(𝑥) = 𝑥
as 𝑥 approaches 𝑐 is equal to 𝑥

ROOT LAW
The limit of the nth root of a function
CONSTANT MULTIPLE LAW is equal to the nth root of the limit of
The limit of a constant k the function, where 𝑛 is a positive
multiplied to f(x) is equal to integer, and the limit of the function
the constant k multiplied to is positive when 𝑛 is even
the limit of f(x)

SUM OR DIFFERENCE LAW

The limit of the sum or
difference of two functions is
equal to the sum or difference
of the limits of the two - END -
functions.
 LIMITS OF ALGEBRAIC
FUNCTIONS
BASIC CALCULUS : WEEK 2.1

The limit of a polynomial EXTRA TIPS!

function 𝒇(𝒙) as 𝑥 approaches 𝑐
is equal to 𝒇(𝒄) RADICALS

The limit of a rational function

as 𝑥 approaches 𝑐 is equal to
provided 𝑄(𝑐) ≠ 0.

The limit of a radical function

FACTORING
where 𝑃(𝑥) is a
polynomial in 𝑥, as 𝑥 approaches 𝑐 is

for any real number c

We can use factoring and

rationalizing technique to
evaluate indeterminate forms
of functions of type
To verify if the function is
indeterminate, evaluate it at 𝑥
=c

when, is in indeterminate
form (zero over zero), we can
factor out n(x) and d(x) to
cancel common factors
As long as the functional
value is not less than
zero/unreal, then you can find
the limit of radical functions
by substituting

- END -
 INFINITE LIMITS AND
LIMITS AT INFINITY
BASIC CALCULUS : WEEK 3

Infinite Limits Limits At Infinity

THEOREM 1
If n is any positive real number If n is any positive real number, then

THEOREM 2

Focus on the Leading Terms of the

numerator and denominator
If the leading terms have similar
exponents (n=m), then the limit is
If n < m, the limit is 0
If n > m, the limit Does Not Exist

- END -
 LIMITS OF SOME
TRANSCENDENTAL
FUNCTIONS
BASIC CALCULUS : WEEK 4

exponential. baka magamit?

Let 𝑎 and 𝑏 be real numbers,
where 𝑏 > 0 and 𝑏 ≠ 1. Then

logarithmic.
Let 𝑎 and 𝑏 be real numbers,
where 𝑎 > 0, 𝑏 > 0, and 𝑏 ≠ 1.
Then

trigonometric.
Let 𝑎 be a real number in the
domain of the given
trigonometric function, and 𝑘
an integer.

MAGIC HEXAGON (TRIG IDENTITY)

https://youtu.be/yzyNJVzByew

special limits.

bat kaya special limits,

siguro may leche flan or ice
cream on top
 CONTINUITY OF A
FUNCTION
BASIC CALCULUS : WEEK 4

A function is continuous if it’s

graph doesn’t have any jumps Continuity of a
or breaks. It also isn’t
continuous if a function
Function at an
extends to infinity Interval
An absolute value function is
Given the equation, a function continuous at any value
is continuous if:
Polynomial functions are
The Functional Value f(c) exists continuous in the set of Real
Numbers
The Limit exists, and the one
sided limits are equal If asked to give the interval
where the rational function is
The Functional Value and continuous, check its domain.
Limits are EQUAL
Create an interval that
If ANY one of these rules are excludes the restriction of the
unsatisfied, then the function domain
is NOT CONTINUOUS

Rational functions are

continuous as long as the
function is still in its domain
Squareroot functions are
continuous as long as x
is NOT less than 0 (not negative)

iiyakan na...ayoko na
mabuhay pls T-T

walang IVT and EVT sa

reviewer na i2 dahil kahit
ako di ko din gets yun
 DEFINITION OF LIMIT AS
DERIVATIVE OF A
FUNCTION
BASIC CALCULUS : WEEK 7

Continuity and
Differentiability of
Functions
A function 𝑓 is continuous at 𝑥 = 𝑐 if
the following three conditions are
satisfied
i. 𝑓(𝑐) is defined;
ii. lim𝑓(𝑥) exists; and
→
→
𝑥 𝑐

If 𝑦 = 𝑓(𝑥), the derivative of 𝑓 is iii. 𝑓 𝑐 = lim 𝑥 𝑐 𝑓(𝑥).

commonly denoted by: If at least one of the conditions fails,
then we say that 𝑓 is discontinuous
at 𝑥 = 𝑐.
A function 𝑓 is differentiable on the
interval (𝑎, 𝑏) if:
exists for every 𝑐 in (𝑎, 𝑏).

Average and
Instantaneous Rate A function 𝑓 is differentiable at 𝑥 = 𝑐 if

of Change 𝑓′(x) exists which means that

𝑓 is continuous at 𝑥 = 𝑐.
The average rate of change of 𝑦 with
respect to 𝑥 on [𝑥₀, 𝑥] CONTINUITY DOES NOT NECESSARILY
IMPLY DIFFERENTIABILITY.
If 𝑓 is not continuous at 𝑥 = 𝑐, then 𝑓 is
not differentiable at 𝑥 = 𝑐

If 𝑓 is not differentiable at 𝑥 = 𝑐, it does

The instantaneous rate of change of 𝑦 not mean that 𝑓 is not continuous at 𝑥
with respect to 𝑥 at 𝑥 = 𝑥₀ =𝑐

A function is not differentiable at 𝑥 = 𝑐

if one of the following is true:
a. 𝑓 is not continuous at 𝑥 = 𝑐.
b. the graph of 𝑓 has a vertical
tangent line at 𝑥 = 𝑐.
c. the graph of 𝑓 has a corner or cusp
at 𝑥 = 𝑐.
 BASIC DIFFERENTIATION
RULES
BASIC CALCULUS : WEEK 8

Exponential
Functions

Logarithmic
Functions

Trigonometric
Functions

Implicit

- END -