UPSC NDA Math Model Paper 9

lim

x→0

tan‌x−x

x2‌tan‌x

=

0

0

is an indeterminate form. Let us simplify and use the L'Hospital's Rule.

lim

x→0

tan‌x−x

x2‌tan‌x

=

lim

x→0

[

tan‌x−x

x3

×

x

tan‌x

]
We know that

lim

x→0

x

tan‌x

=1, but

lim

x→0

tan‌x−x

x3

is still an indeterminate form, so we use L'Hospital's Rule:

lim

x→0

tan‌x−x

x3

=

lim

x→0

sec2x−1

3x2

, which is still an indeterminate form, so we use L'Hospital's Rule again:

lim

x→0

sec2x−1

3x2

=

lim

x→0

2‌sec‌x(sec‌x‌tan‌x)

6x

=

lim

x→0

sec2x‌tan‌x

3x

, which is still an indeterminate form, so we use L'Hospital's Rule again:

lim

x→0

sec2x‌tan‌x

3x

=

lim

x→0

sec2xsec2x+tan‌x[2‌sec‌x(sec‌x‌tan‌x)]

3

=

1

3

∴

lim

x→0

tan‌x−x

x2‌tan‌x

=1×

1

3

=

1

3

.