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QUESTION: 1

is equal to

Solution:

lt h->0 [sin(x+h)^{½} - sin(x)^{½}]/h

Differentiate it with ‘h’

lt h->0 {cos(x+h)^{½} * ½(x+h)^{1/2}] - 0}/1

lt h->0 {cos(x+h)^{½} * ½(x+h)^{1/2]}

lt h->0 cos(x)^{½} / 2(x)^{½}

QUESTION: 2

Solution:

Divide the numerator and denominator by x, so that given function becomes

f(x)=1+sinx/x/(1+cosx/x)

Now as x→∞.sinx/x→0 , because sin x would oscillate between +1 and -1, which in either case divided by ∞ would be 0. Thus the limit of the numerator would be 1. Like wise the limit of the denominator would also be 1.

Thus limit as a whole would be 1

QUESTION: 3

Solution:

QUESTION: 4

is equal to

Solution:

lim x→0 sin x^{n} . (x)^{m} . x^{n} /(sin x)^{m}.(x)^{m}.x^{n}

lim x→0 sin x^{n}.(x)^{m}.x^{n-m}/x^{n}.(sin x)^{m}

Applying limits.

=0^{n-m} = 0

QUESTION: 5

Solution:

QUESTION: 6

If G(x) = then has the value

Solution:

QUESTION: 7

then dy/dx is equal to

Solution:

QUESTION: 8

If f be a function such that f (9) = 9 and f ‘ (9) = 3, then is equal to

Solution:

QUESTION: 9

is equal to

Solution:

QUESTION: 10

If f(x) = , x ∈ (0,1), then f'(x) is equal to

Solution:

QUESTION: 11

If y = sin-11 x and z = cos -1 then dy/dz =

Solution:

QUESTION: 12

is equal to

Solution:

QUESTION: 13

The function, f(x) = and f(a) = 0, is

Solution:

QUESTION: 14

is equal to

Solution:

QUESTION: 15

is equal to

Solution:

QUESTION: 16

If sin x = then dx/dy is equal to

Solution:

QUESTION: 17

holds true for

Solution:

QUESTION: 18

Dervative of tan w.r.t is

Solution:

QUESTION: 19

If y = log then

Solution:

QUESTION: 20

If y = log x , then y_{n} =

Solution:

QUESTION: 21

The derivative of sec^{-1} with respect to at x = 1/x is

Solution:

QUESTION: 22

is equal to

Solution:

QUESTION: 23

is eqaual to

Solution:

QUESTION: 24

If y = then dy/dx =

Solution:

QUESTION: 25

then at x = 1, f(x) is

Solution:

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