Post on 05-Apr-2018
8/2/2019 Limits Theory
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Prof. Sunil Patils TutorialsXII - Mathematics
1 L I M I T S A N D C O N T I N U I T Y
L I M I T S
Definition : We say that l x f a x = )(lim given any > 0, we can say find >0 such that ,| f(x) l | < where 0 < | x a | <
Standard Limits :
1) 1lim
= nnn
a xan
a xa x
( a > 0 n Q )
2) 1sin
lim0
= x
x x
; 1sin
lim0
= kx
kx x
; 1)(
sinlim
0=
n
n
x kx
kx; 1
tanlim
0=
x x
x
3) a x
a x
xlog
1lim
0=
; a
xk
a xk
xlog
1lim
0=
4) 11
lim0
= x
e x
x; 1log
1lim
0==
e
xk e xk
x
5) e x x x
=+
1
0)1(lim ; e xk xk
x=+
1
0)1(lim
6) 1log)1log(
lim0
==+
e x
x x
; 1log)1log(
lim0
==+
ekx
xk x
7) 01
lim = x x
; 01
lim = k x x
( k > 0)
Algebra of limits :
1) [ ] )(lim)(lim)()(lim x g x f x g x f a xa xa x =
2) [ ] )(lim)(lim x f k xkf a xa x =
3) [ ] )(lim)(lim)()(lim x g x f x g x f a xa xa x =
4) [ ] )(lim)(lim)()(lim x g x f x g x f a xa xa x
=
5) If f(x) < g(x) then )(lim)(lim x g x f a xa x
6))(lim
)( )()]([lim x g
a x
x g
a x
a x
x f iml x f =
Limits and Continuity -Theory -1
8/2/2019 Limits Theory
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Prof. Sunil Patils TutorialsXII - Mathematics
C O N T I N U I T Y1) A function f is said to continuous at x = a if
f(a) exists , )(lim x f a x + exists ,
)(lim x f a x exists and
)(lim x f a x + = )(lim x f a x = f(a)
2) A function f is said to continuous at x = a if f(a) exists , )(lim x f a x exists
then )(lim x f a x = f(a)
3) Removable Discontinuity :
If f(x) is such that : )(lim x f a x exists and)(lim x f
a x f(a)
then f has a removable discontinuity at x = a and this discontinuity can be removed
by redefining the function suitably at x = a
4) Continuity in an interval :
A function f is said to continuous on [ a, b ] if
i) f is continuous at every c ( a, b)
i.e. )(lim x f c x + =
)(lim x f c x = f(c)
ii) f is continuous from the right at x = a and continuous from the left the left at x = b
i.e. )(lim x f a x
+
= f(a) and )(lim x f
b x
= f(b)
5) Continuity of Standard function :
i) A polynomial is continuous for all real values of x.
If nn xa xa xaa x f ++++= .........)(2
210 where N n and a 0 , a 1 , a 2 , an
are real constants then f is said to be a polynomial of degree n.
ii) f(x) = sin x and f(x) = cos x are continuous for all real values of x
6) Algebra of Continuous Function:
If f and g are both continuous at x = c then
i) f g , ii) f g , iii) f g [ provided g(c) 0 ] are continuous at x = c
Limits and Continuity -Theory -2