Limits, Continuity
& Differentiability
Limit
Let y = f ( x ) be a function of x. If at x = a,f ( x ) takes indeterminate form
0 ∞ ∞ 0
, , ∞ − ∞ , 0 × ∞ , 1 , 0 and ∞ , then we consider the values of the
0
0 ∞
function at the points which are very near to a. If these values tend to
a definite unique number as x tends to a, then the unique number, so
obtained is called the limit of f ( x ) at x = a and we write it as lim f ( x ).
x→a
Left Hand and Right Hand Limits
If values of the function, at the points which are very near to the left of
a, tends to a definite unique number, then the unique number so
obtained is called the left hand limit of f ( x ) at x = a. We write it as
f ( a − 0) = lim f ( x ) = lim f ( a − h )
x → a− h → 0+
Similarly, right hand limit is written as
f ( a + 0) = lim f ( x ) = lim f ( a + h )
x → a+ h → 0+
Existence of Limit
lim f ( x ) exists, if
x→a
(i) lim f ( x ) and lim f ( x ) both exist
x → a− x → a+
(ii) lim f ( x ) = lim f ( x )
x → a− x → a+
Uniqueness of Limit
If lim f ( x ) exists, then it is unique, i.e. there cannot be two distinct
x→a
numbers l1 and l2 such that when x tends to a, the function f ( x ) tends
to both l1 and l2.
, Fundamental Theorems on Limits
If f ( x ) and g( x ) are two functions of x such that lim f ( x ) and lim g( x )
x→a x→a
both exist, then
(i) lim [ f ( x ) ± g( x )] = lim f ( x ) ± lim g( x )
x→a x→a x→a
(ii) lim [kf ( x )] = k lim f ( x ), where k is a fixed real number.
x→a x→a
(iii) lim [ f ( x ) g( x )] = lim f ( x ) lim g( x )
x→a x→a x→a
lim f ( x )
f(x) x → a
(iv) lim = , provided lim g( x ) ≠ 0
x→a g( x ) lim g( x ) x→a
x→a
lim g ( x )
(v) lim [ f ( x )] g ( x ) = lim f ( x )
x→ a
x→a x → a
(vi) lim ( gof ) ( x ) = lim g[ f ( x )] = g lim f ( x )
x→a x→a x → a
(vii) lim log f ( x ) = log lim f ( x ) , provided lim f ( x ) > 0.
x→a x → a x→a
lim f ( x )
(viii) lim e f ( x ) = ex → a
x→a
(ix) If f ( x ) ≤ g( x ) for every x excluding a, then lim f ( x ) ≤ lim g( x ).
x→a x→a
(x) lim f ( x ) = lim f ( x )
x→a x→a
1
(xi) If lim f ( x ) = + ∞ or − ∞ , then lim =0
x→a x→a f(x)
Important Results on Limits
1. Algebraic Limits
xn − an
(i) lim = na n −1, n ∈ Q
x→ a x−a
(1 + x )n − 1
(ii) lim = n , n ∈Q
x→ 0 x
& Differentiability
Limit
Let y = f ( x ) be a function of x. If at x = a,f ( x ) takes indeterminate form
0 ∞ ∞ 0
, , ∞ − ∞ , 0 × ∞ , 1 , 0 and ∞ , then we consider the values of the
0
0 ∞
function at the points which are very near to a. If these values tend to
a definite unique number as x tends to a, then the unique number, so
obtained is called the limit of f ( x ) at x = a and we write it as lim f ( x ).
x→a
Left Hand and Right Hand Limits
If values of the function, at the points which are very near to the left of
a, tends to a definite unique number, then the unique number so
obtained is called the left hand limit of f ( x ) at x = a. We write it as
f ( a − 0) = lim f ( x ) = lim f ( a − h )
x → a− h → 0+
Similarly, right hand limit is written as
f ( a + 0) = lim f ( x ) = lim f ( a + h )
x → a+ h → 0+
Existence of Limit
lim f ( x ) exists, if
x→a
(i) lim f ( x ) and lim f ( x ) both exist
x → a− x → a+
(ii) lim f ( x ) = lim f ( x )
x → a− x → a+
Uniqueness of Limit
If lim f ( x ) exists, then it is unique, i.e. there cannot be two distinct
x→a
numbers l1 and l2 such that when x tends to a, the function f ( x ) tends
to both l1 and l2.
, Fundamental Theorems on Limits
If f ( x ) and g( x ) are two functions of x such that lim f ( x ) and lim g( x )
x→a x→a
both exist, then
(i) lim [ f ( x ) ± g( x )] = lim f ( x ) ± lim g( x )
x→a x→a x→a
(ii) lim [kf ( x )] = k lim f ( x ), where k is a fixed real number.
x→a x→a
(iii) lim [ f ( x ) g( x )] = lim f ( x ) lim g( x )
x→a x→a x→a
lim f ( x )
f(x) x → a
(iv) lim = , provided lim g( x ) ≠ 0
x→a g( x ) lim g( x ) x→a
x→a
lim g ( x )
(v) lim [ f ( x )] g ( x ) = lim f ( x )
x→ a
x→a x → a
(vi) lim ( gof ) ( x ) = lim g[ f ( x )] = g lim f ( x )
x→a x→a x → a
(vii) lim log f ( x ) = log lim f ( x ) , provided lim f ( x ) > 0.
x→a x → a x→a
lim f ( x )
(viii) lim e f ( x ) = ex → a
x→a
(ix) If f ( x ) ≤ g( x ) for every x excluding a, then lim f ( x ) ≤ lim g( x ).
x→a x→a
(x) lim f ( x ) = lim f ( x )
x→a x→a
1
(xi) If lim f ( x ) = + ∞ or − ∞ , then lim =0
x→a x→a f(x)
Important Results on Limits
1. Algebraic Limits
xn − an
(i) lim = na n −1, n ∈ Q
x→ a x−a
(1 + x )n − 1
(ii) lim = n , n ∈Q
x→ 0 x
