This ppt covers following topic of unit - 1 of B.Sc. 1 Calculus :- Definition of limit , left & right hand limit and its example , continuity & its related example.

1. Shri Shankaracharya Mahavidyalaya
JUNWANI,BHILAI NAGAR (C.G)
TOPIC:-LIMIT AND CONTINUITY
Presented by:
Mrs. Preeti Shrivastava

2. CONTENT
(1) DEFINATION OF LIMIT.
(2) DEFINACTION OF LEFT HAND
LIMIT.
(3) DEFINACTION OF RIGHT HAND
LIMIT.
(4) EXAMPLE.
(5) CONTINUITY.
(6) SOME EXAMPLES.

3. LIMITS
DEFINACTION:- Let f(x) be a function
of the variable x, then the value
of f(x) at x=a, i.e. for some value
`a’ of x is denoted by f(a). The
value of f(a) may be of two kinds.

4. (i) A definite and finite quantity:-
In this case the value of f(x) is said
to be defined at x=a.
(ii) An indefinite and indeterminate
quantity:-
In this case the value of f(x) is said
to be undefined at x=a.

5. LEFT HAND LIMIT
DEFINITION:- A function f(x) is sadi to
tend to `l’ as a tends to a through the
values less than a (or from left), if for
any given there exists a such
that.
i.e. for every f(x)
0 0
 ,,aax     ll ,
  lxfaxa )(

6. Symbolicelly, we write:-
Or
Or f(a-0)=
And is called the left hand limit (L.H.L.)
lxf
ax


)(lim0
lxf
ax


)(lim
l

7. RIGHT HEND LIMIT
DEFINITION:- A function f(x) is said to be tend to
limit l as x tends to a, through the values
greater than a (or form right), if for every
there exist a such that
i.e. for every
Symbolically, we write:-
0
0
  lxfaxa )(
),()(),,(   llxfaax
0)(lim0


xf
ax

8. Or f(a+0) =
And is called the right hand limit (R.H.L.)
Ex:- If
F(x)= when x<1
when x>1
Find if it exist.
lxf
ax


)(lim
 ,23 x
,34
2
xx 
),(lim1
xf
x
l

9. SOLUCTION:- consider the L.H.L.
)1()( limlim 01
hfxf
hx


 
 
0
01
0.31
)31(
233
2)1(3
lim
lim
lim
0
0
0









h
h
h
h
h
h

10. Again, consider R.H.L.
)1(limlim 01
hf
hx

 
 )1(3)1(4 2
0
lim hh
h


 11  h
 
 
 
1
10.50.4
154
33844
33)21(4
2
2
0
2
0
2
0
lim
lim
lim








hh
hhh
hhh
h
h
h

11. 1)()( limlim 11



xfxf
xx
hence
1)(lim1


xf
x

12. CONTINUITY:-
The intuitive concept of continuity of a
function is derived from its geometrical
construction. If the graph of the function
y=f(x) is curve which does not break at the
point x=a then the function y=f(x) is called
continuous at x=a.
If the graph of the function break at some
point then this point is called the point of
discontinuity.

13. Ex:- IF
F(x)= x -1
, x=-1
Is f(x) continuous at x=-1?
 ,
1
12


x
x
2


14. Soluction L:- R.H.L. at x=-1
F(-1+0)
2
20
2
)2(
2
11
)121(
1)1(
1)1(
)1(
lim
lim
lim
lim
lim
lim
0
0
2
0
2
0
2
0



















h
h
h
hh
h
hh
h
h
hf
h
h
h
h
oh
h

15. L.H.L. at x= -1
F(-1-0)
2
20
)2(
2
11
121
1)1(
1)1(
)1(
lim
lim
lim
lim
lim
0
2
0
2
0
2
0
0


















h
h
hh
h
hh
h
h
hf
h
h
h
h
h

16. Again when x=-1, then f(x)= -2
f(x)=2
Since f(-1-0)= f(-1)
Hence the given function is continuous at x= -1


17. The following function for continuity at the
origin.
f(x)= , if x 0
, if x = 0
SOLUCTION:- Here f(0)=0
R.H.L. f(0+0)


















)(
11
)0(
1)0(
lim
)0(lim
0
0
ho
e
h
eh
hf
h
h
0
1
1
1
x
x
e
xe


18. 0
10
0
1
0
1
0
1
1
1
1
0
1
1
0
1
1
1
0
1
1
0
lim
lim
lim



































e
e
e
h
e
e
he
e
he
hh
h
h
h
h
h
h
h

19. L.H.L. = F(0-0)
since f(0+0) = f(0),so f(x) is continuous at x=0.
0
01
0
1
.0
1
1
)0(
)0(
0
1
0
1
1
1
0
)0(
1
)0(
1
0
0
lim
lim
lim





























e
e
e
he
e
eh
hf
h
h
h
h
h
h
h

20. THANK YOU