UPSC NDA Math Model Paper 6

lim

x→ π 4

[

1−sin‌2‌x

1+cos‌4‌x

]=

0

0

Since this is an indeterminate form, we can apply the L-Hospital Rule as:

lim

x→a

f(x)

g(x)

=

lim

x→a

f′(x)

g′(x)

lim

x→ π 4

[

1−sin‌2‌x

1+cos‌4‌x

] =

lim

x→ π 4

[

−2‌cos‌2‌x

−4‌sin‌4‌x

]=

0

0

Since this is also 0/0 form, we again apply the L-Hospital Rule as:

lim

x→ π 4

[

−2‌cos‌2‌x

−4‌sin‌4‌x

] =

lim

x→ π 4

[

4‌sin‌2‌x

−16‌cos‌4‌x

]=

1

4

Hence, option A is the correct answer.