UPSC NDA Math Model Paper 9

Since the expression for f(x) is same for x < 1 and x > 1, we have:

lim

x→1+

f(x)=

lim

x→1−

f(x)=

lim

x→1

f(x) =

lim

x→1

logex

x−1

which is an indeterminate form.
By using the L'Hospital's rule:

lim

x→1

logex

x−1

=

lim

x→1

1 x

1

=1
Since, f(x) is continuous at x=1, we have:

lim

x→1

f(x)=f(1)
⇒1=k
Hence, the value of k is 1 .