UPSC NDA Math Model Paper 7

Here, we have to find the limit of

lim

x→0

ax−bx

ex−1

The expression

ax−bx

ex−1

can be re-written as:
⇒

ax−bx

ex−1

=[

ax−1 x − bx−1 x

ex−1 x

]
Now by applying limits on both the sides of the above equation we get
⇒

lim

x→0

ax−bx

ex−1

=

lim

x→0

[

ax−1 x − bx−1 x

ex−1 x

]
As we know that,

lim

x→a

[

f(x)

g(x)

]=x→a

lim

x→a

g(x), provided

lim

x→a

g(x)≠0
⇒

lim

x→0

[

ax−1 x − bx−1 x

ex−1 x

] =

[ lim x→0 ax−1 x ]−[ lim x→0 bx−1 x ]

[ lim x→0 ex−1 x ]

As we know that,

lim

x→0

[

ax−1

x

]=log‌a,a>0 and

lim

x→0

[

ex−1

x

]=1
⇒

lim

x→0

ax−bx

ex−1

=log‌

a

b

Hence, option A is true.