If x → c we know that h → 0 Therefore
lim
( )
x
c f x
→
 = lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→  f (x) = f(c) and hence f is a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function