TS EAMCET 4 May 2018 Shift 1 Solved Paper

Consider the function.
f (x) = {

x2‌log(cosx) log(1+x2)	,x≠0
0	,x=0

LHL at x=0 is,

lim

h→0

log(cos‌x)

log(1+x2) x2

=

lim

h→0

log(cos−h)

log(1+h2) h2

=

log(1)

1

=0
RHL at x=0 is,

lim

h→0

log(cos‌x)

log(1+x2) x2

=

lim

h→0

log(cos‌0−h)

log(1+h2) h2

=

log(1)

1

=0
Therefore, the function f(x) is continuous at x=0.
Now, LHD at x=0 is,

lim

h→0

(0−h)2‌log(cos(0−h)) log(1+h2) −0

−h

=

lim

h→0

(h)2‌log(cos(h)) log(1+h2)

−h

=

lim

h→0

h× log(cos(h)) log(1+h2)

−h2

=0×

log(1)

−1

=0
Now, RHD at x=0 is,

lim

h→0

(0+h)2‌log(cos(0+h)) log(1+h2) −0

h

=

lim

h→0

(h)2‌log(cos(h)) log(1+h2)

h

=

lim

h→0

h× log(cos(h)) log(1+h2)

h2

=0×

log(1)

−1

=0
Therefore, the function is differentiable at x=0.