Let [x] denote the greatest integer not exceeding x, If l_1 = \lim_{x\to2^{\+}} (x^2+[x]) , l_2 = \lim_{x\to3^{-}} (2x - [x]) and l_3 = \lim_{x\to\pi/2} (\frac{cosx}{x-\pi/2}) then

Correct Answer is : l3 < l2 < l1

AP EAMCET Engineering 23 Apr 2019 Shift 1 Paper

Section: Mathematics

Question : 62 of 160

Marks: +1, -0

Let [x] denote the greatest integer not exceeding x,
If l1 =

lim

x→2+

(x2+[x]) , l2 =

lim

x→3−

(2x - [x]) and l3 =

lim

x→

(

cosx

x−

) then

l2 < l3 < l1
l1 < l3 < l2
l1 < l2 < l3
l3 < l2 < l1

Validate

Solution:

Consider the given expression,
l1=

lim

x→2+

(x2+[x])
=

lim

x→2+

(x2+2)
=4+2
=6
And,
l2=

lim

x→3−

(2x−[x])
=

lim

x→3−

(2x−2)
=2(3)−2
=4
And,
13=

lim

x→

(

cos‌x

x−

)
Take x−

=y then,
l3=

lim

y→0

(

cos(

+y)

)
=

lim

y→0

(−

sin‌y

)
=−1
This implies,
$l_{3}

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