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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

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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