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LIMITS
Logarithmic Function
The logarithmic form of
x = b
y
is
log
b
x = y .
Given that b>0,
lim
x→ + ∞
log
b
x
is + ∞
Given that b>0,
lim
x→ 0
+¿
log
b
x ¿
is - ∞
lim
x→ + ∞
ln x
is + ∞
lim
x→ 0
+¿
ln 4 xis − ∞ ¿
A vertical asymptote is a vertical line that
the graph of the function approaches as f (x)
approaches positive infinity or negative
infinity.
Note: Logarithmic functions have vertical
asymptotes in which it is the restriction of
the function.
Exponential Function
Functions having an exponent that is/has a
variable and whose base is greater than
zero and is not equal to 1 is called
exponential function.
Given that b>0,
lim
x→ + ∞
b
x
is + ∞
Given that b>0,
lim
x→ − ∞
b
x
is 0
Given that 0<b<1,
lim
x→ + ∞
b
x
is 0
Given that 0<b<1,
lim
x→ − ∞
b
x
is + ∞
The
lim
x→ + ∞
e
x
is + ∞
The
lim
x→ − ∞
e
x
is 0
The
lim
x→ + ∞
e
− x
is 0.
The
lim
x→ − ∞
e
− x
is + ∞
A horizontal asymptote is a horizontal line
that the graph of the function approaches as
x approaches positive infinity or negative
infinity.
Note: An exponential function has a
horizontal asymptote if the limit exists.
Whatever the limit of the function if it exists
is automatically the horizontal asymptote of
the function.
Rational functions have either horizontal or
vertical asymptote and some have both.
Trigonometric Functions
The
lim
x → 0
sin t
t
is 1.
The
lim
x → 0
1 −cos t
t
is 0.
The lim
x → 0
e
t
t
is 1.
CONTINUITY
A function f ( x ) is continuous at a number c
if all of the following conditions are satisfied:
a. f ( c ) exists
b.
lim
x →c
f ( x )
exists
c.
f ( c ) =lim
x→ c
f ( x )
DISCONTINUITY
If one of the three conditions was not
satisfied, then the function is discontinuous.
Types of Discontinuity:
a. Removable Discontinuity - it occurs
when there is a hole in the graph of a
function.
Note: We can redefine the function to
remove the discontinuity.
Example: f
x
2 x
2
x + 3
, the redefined
function is:
f ( x )=
2 x
2
x + 3
; x ≠ − 3
− 7 ; x =− 3
Steps: 1. Copy the original function then x ≠ should not
be equal to the restriction.
- Find the limit then x = restriction
b. Jump Discontinuity - it occurs when the
graph of the function stops at one point and
seems to jump at another point. In a jump
discontinuity, the left hand and the right
hand limits exist but are not equal.
c. Infinite Discontinuity - it occurs when the
function has at least one infinite limit.
Intermediate Value Theorem
Intermediate Value theorem states that if the
function f ( x )
is continuous on the closed interval
[a,b] from point A to point B, then the curve crosses
every line
y = y
3
in at least one point and this line
lies between the line
y = y
1
and
y = y
2
In other words a function f ( x ) which is found to be
continuous over a closed interval [ a , b ] will take
any value between f ( a ) and f ( b ).
Extreme Value Theorem
The figure illustrates The Extreme Value Theorem
which states that a function f ( x )
which is found to
be continuous over a closed interval [a,b] is
guaranteed to have extreme values in that interval.
An extreme value of
f , or extremum, is either a
minimum or a maximum value of the function. A
minimum value of f occurs at some x = c if
f ( c ) ≤ f ( x )
for all x ≠ c in the interval. A maximum
value of f occurs at some x = c if f ( c ) ≤ f ( x )x for all
x ≠ c in the interval.
TANGENT LINE
The tangent lines at the “peaks” and
“troughs” of a smooth curve are horizontal.
The tangent line drawn on ( 0 , ± 1 ) and ( ± 1,0)
is shown below
At ( ± 1,0), the tangent lines are vertical and
at ( 0 , ± 1 ) , the tangent lines are horizontal.
The tangent line drawn on the points at the
first, second, third and fourth quadrant is
shown below
At points in the first and third quadrants, the
tangent lines are slanting to the left and at
points in the second and fourth quadrants,
the tangent lines are slanting to the right.
As Q approaches point P, then PQ will
be tangent to the graph.
The tangent line to y = f ( x )at point P is
the limiting position of all secant lines
PQ as point Q approaches point P.
Equation of the Tangent Line
The steps in finding the equation of the tangent line
of any function are:
Find the value of
x
0 ,
y
0
.
Get the slope of the tangent line by
computing
m = lim
x→ x
0
y − y
0
x − x
0
Substitute this value of m and the
coordinates of the known point
P
x
0 ,
y
0
)into
the point-slope form denoted by
y − y
0
= m ( x − x
0
Example:
Let us find the equation of the tangent line of
y = x
2
at x = 2
.
Find the value of
x
0 ,
y
0
.
x
0
To find the value of
y
0
by substituting
x
0
into
y = x
2
.
y
0
2
Get the slope of the tangent line.
m =lim
x → 2
x
2
x − 2
Substitute this value of m and the
coordinates of the known point
P
x
0 ,
y
0
into
the point-slope form.
y − 4 = 4 ( x − 2 )
y = 4 x − 4
DERIVATIVE
Let the graph below be the graph of a
function f ( x )with P ( x , f ( x )) as the
coordinate. Let us locate another point on
the graph which will be labeled as point Q
with coordinate ( x + ∆ x , f ( x + ∆ x )) and
connect the two lines with a secant line PQ.
f
'
( x )=( 3 x
2
− 5 x + 1 )
dy
dx
( 3 x
2
− 4 ) + ( 3 x
2
dy
dx
( 3 x
2
− 5 x + 1 )
3 x
2
− 5 x + 1
6 x
3 x
2
3 x − 5
¿ 18 x
3
− 30 x
2
3
− 15 x
2
− 12 x + 20
¿ 27 x
3
− 45 x
2
− 6 x + 2 0
Quotient Rule
If f ( x )=
g ( x )
h ( x )
are differentiable
functions and g ( x ) ≠ 0 , then f ' ( x )= h ( x )¿ ¿
.
f ' ( x )=
( 5 x
2
− 3 x + 2 )
dy
dx
( 3 x
2
+ 4 )−( 3 x
2
dy
dx
( 5 x
2
− 3 x + 2 )
( 5 x
2
− 3 x + 2 )
2
=
( 5 x
2
− 3 x + 2 )( 6 x )−
3 x
2
10 x − 3
5 x
2
− 3 x + 2
2
=
30 x
3
2
− 12 x − 30 x
3
2
− 40 x + 12
( 5 x
2
− 3 x + 2 )
2
=
27 x
2
− 52 x + 12
( 5 x
2
− 3 x + 2 )
2
Chain Rule
It is in the form f ( g ( x ) )= f
'
( g ( x ) )∗ g ( x )
f ( x )=( 15 x
2
− 6 x + 2 )
1
2
f ' ( x )=
( 15 x
2
− 6 x + 2 )
1
2
− 1
dy
dx
( 15 x
2
− 6 x + 2 )
( 15 x
2
− 6 x + 2 )
− 1
2
( 30 x − 6 )
(
30 x − 6
)
( 15 x
2
− 6 x + 2 )
− 1
2
15 x − 3
√
15 x
2
− 6 x + 2
f ( x )=( 3 x − 2 )( 5 x + 4 )
2
f ' ¿
¿ ( 3 x − 2 ) ( 2 ) ( 5 x + 4 )
2 − 1
( 5 )+( 5 x + 4 )
2
¿ 10 ( 3 x − 2 )( 5 x + 4 )+ 3 ( 5 x + 4 )
2
¿ 10 ( 15 x
2
2
¿ 150 x
2
2
¿ 225 x
2
DIFFERENTIABILITY AND CONTINUITY
If f ( x ) is continuous at x = a , it does not
mean that ( x )
is di erentiable atff x = a.
If f ( x ) is not continuous at x = a , then f is
not di erentiableff x = a.
If f ( x )
is not di erentiable atff x = a , it does
not mean that f ( x ) is not continuous at
x = a.
A function f ( x ) is not di erentiable atff x = a
if one of the following is true:
a. f ( x ) is not continuous at x = a.
b. the graph of f ( x ) has a vertical
tangent line at x = a
c. the graph of f ( x ) has a corner or
cusp at x = a.
