UPSC NDA Math Model Paper 10

Here, we have to find the value of

lim

x→0

[

√5+x−√5−x

x

]
Let f(x)=

√5+x−√5−x

x

As we can see that the numerator of f(x) contains surds so we need to simplify the f(x) by using its conjugate surds.
Here the conjugate surds of the numerator of f(x) is: √5+x−√5−x
So, by multiplying both the numerator and denominator of f(x) with √5+x−√5−x we get,
⇒f(x)=

2x

x.(√5+x+√5−x)

Now, lets evaluate the limit of f(x) as x→0,
⇒

lim

x→0

f(x)=

lim

x→0

[

√5+x−√5−x

x

] =

1

√5