Home
Class 12
MATHS
Let f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),...

Let `f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),(0,if,x=0):}` (where [x] denotes the greatest integer function). Then the correct statement is/are

A

Limit exists for `x=-1`.

B

f(x) has a removable discontinuity at x = 1.

C

f(x) has a non removable discontinuity at x = 2.

D

f(x) is discontinuous at all positive integers.

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`f(1^(+))=underset(xrarr1^(+))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr1^(+))(lim)(x(0)+x(1))`
`=1`
`f(1^(-))=underset(xrarr1^(-))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr1^(-))(lim)(x(0)+x(1))`
`=1`
`f(2^(+))=underset(xrarr2^(+))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr2^(+))(lim)(x(0)+x(2))`
`=4`
`f(2^(-))=underset(xrarr2^(-))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr2^(-))(lim)(x(0)+x(2))`
`=2`
Obviously f(x) is discontinuous at all positive integers but at x = 1 it has removable discontinuity.
Promotional Banner

Topper's Solved these Questions

  • CONIC SECTIONS

    CENGAGE PUBLICATION|Exercise All Questions|102 Videos

  • COORDINATE SYSTEM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|2 Videos

Similar Questions

Explore conceptually related problems

Solve x^2-4-[x]=0 (where [] denotes the greatest integer function).

Let f(x) =x[1/x]+x[x] if x!=0 ; 0 if x =0 where[x] denotes the greatest integer function. then the correct statements are (A) Limit exists for x=-1 (B) f(x) has removable discontinuity at x =1 (C) f(x) has non removable discontinuity at x =2 (D) f(x) is non removable discontinuous at all positive integers

If f(x)= [sin^2x] (where [.] denotes the greatest integer function ) then :

lim_(xrarr1([x]+[x]) , (where [.] denotes the greatest integer function )

If [x]^2-5[x]+6=0 (where [.] denotes the greatest integer function), then x belongs to

If f(x)={(x[x], 0lexlt2),((x-1)[x], 2lexle3'):} , where [.] denotes the greatest integer function, then

Consider the function f(x)={{:(x-[x]-(1)/(2),x !inI),(0, "x inI):} where [.] denotes the greatest integer function and I is the set of integers. Then find g(x)max.[x^(2),f(x),|x|},-2lexle2.

If f(x)=[x], where [.] denotes greatest integer function. Then check the continuity on [1,2]

[x] denote the greatest integer function. The value of int_(1)^(a)[x]f^(')(x)dx,agt1 is-

Solve [cot^(-1) x] + [cos^(-1) x] =0 , where [.] denotes the greatest integer function