Mathematics Times – July 2019

likewise option (b) is true

i.e.,

3

(4) cos

6 2

f



 

and option (c) is

limcosn n 2 1





 

  

 

finally option (d) is

tan cos cos^1 2 1 1 2

8



  

 
      
 
^2 2 1 0 
5.Sol: In all four options, the given function

f x vanishes at x 0. So the question is to

check the behaviour of

f x 

x as x^0 for

Property 1 and of

 

2

f x

x

for Property 2.

Property 1 will hold if f x  is smaller than or

comparable to x, asx 0 In (A), the

ratio

f x 

x equals x which tends to 0

asx 0. In (B) this ratio is

2 1

x3 2. As 2 1

3 2

 ,

the exponent is positive and hence the ratio

tends to 0. So both (A) and (B) are true.

In (C), the ratio equals

x

x

which is 1 for

x 0 and -1 for x 0. So, although both

 

limx 0 2

f x

 x

and

 

limx 0 2

f x

 x

exist, they are

unequal and so

 

limx 0 2

f x

 x

does not exist.

Hence (C) is false.

In (D), the ratio equals 2

sinx sinx 1

x x x

 . The

first factor tends to as x 0. But the second

factor does not tend to any limit. So (D) is

false too.

6.Sol: In both the numerator and the denominator,

we have a sum of the functional values of some

function at the points 1,2, ,n. If we divide

these sums by suitable powers of n, they can

be expressed as the Riemann sums of suitable

functions over the interval 0,1 partitioned

into n equal parts.

For this we factor the coefficient

5.Sol:

6.Sol:

n7 3 in the

denominator as

2 1

n n^3. Then the expression

whose limit is to be taken can be written as

n

n

A

B where

(^13)
1
( )
n
n
k
k
A
 n

 (1)

and  

2

2

1

n

n k

n

B

 an k







2

1

n 1

k k

a

n





  

 

 



(2)

If we multiply both the numerator and the

denominator by

1

n

, then the numerator is a

Riemann sum of  

(^13)
f x x and the
denominator a Riemann sum of the function
 
 
2
1
g x
a x

 , both over the interval 0,1
partitioned into equal parts. Therefor the given
limit equals
1
2
I
I where