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If `f(x) ={([x]^2+sin[x])/underset(0)([x])underset("for"[x]=0) "for" [x]ne 0` where `[x]` denotes the greatest integer function, then , `lim_(xto0)f(x)`, is

A

1

B

0

C

-1

D

non-existent

Text Solution

Verified by Experts

The correct Answer is:
D
We have ,
`lim_(xto0^-)[x]=-1and lim_(xto0^+)[x]=0`
`therefore lim_(xto0^-)f(x)=lim_(xto0^-)((-1)^2+sin(-1))/((-1))=-1+sin1`
and, `lim_(xto0^+)f(x)=lim_(xto0^+)[because f(x)=0 "for"0lex lt 1]`
So, `lim_(x to 0)f(x)` does not exist.
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OBJECTIVE RD SHARMA-LIMITS-Chapter Test
  1. If f(x) ={([x]^2+sin[x])/underset(0)([x])underset("for"[x]=0) "for" [...

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  2. Let f(x)={(x^2,x in Z),((k(x^2-4))/(2-x),x notinZ):} Then, lim(xrarr2...

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  3. If Sn=sum(k=1)^n ak and lim(n->oo)an=a , then lim(n->oo)(S(n+1)-Sn)/sq...

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  4. If a1=1a n da(n+1)=(4+3an)/(3+2an),ngeq1,a n dif("lim")(nvecoo)an=a , ...

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  5. If x1=3 and x +1= sqrt(2+xn), n ge 1, then lim(nto oo) xn is equal to

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  6. The value of lim(xrarr0) (sqrt(1-cosx^2))/(1-cosx), is

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  7. The value of lim(xrarr oo) ((pi)/(4x))sin ((pi)/(4x)), is

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  8. Evaluate ("lim")(nvecoo){cos(x/2)cos(x/4)cos(x/8) cos(x/(2^n))}

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  9. If f(x) is the integral of (2 sin x-sin 2 x)/(x^(3)), "where x" ne 0, ...

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  10. The value of lim(x->0^+) x^m (logx)^n, m,n in N is

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  11. The value of lim(xrarroo) (logx)/(x^n), n gt 0, is

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  12. The value of lim(x->a)log(x-a)/(log(e^x-e^a) is

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  13. Let < an > be a sequence such that lim(x->oo)an=0. Then lim(n->oo)(a1...

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  14. Let lt an gt be a sequence such that a1=1 and an+1 =cos an, n gt 1 . ...

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  15. If f(a) =2, f'(a)=1, g(a) =-1 , g' (a)=2, then the value of lim(xrarr ...

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  16. If f(9)=9,f^(prime)(9)=4,t h e n("lim")(nvecoo)(sqrt(f(x)-3))/(sqrt(x-...

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  17. If Al=(x-ai)/(|x-ai|)30, i=1,2,..., n and if a1 lt a2 lt a3lt ..... lt...

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  18. lim(x -> oo) x^n / e^x = 0, (n is an integer) for

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  19. lim(xrarr0) (x)/(tan^-1x) is equal to

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  20. If f(x) =x , x<0 and f(x)=1 , x = 0, and f(x)=x^2,x>0 then lim(x->0) ...

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  21. lim(xrarroo) sqrt((x+sinx)/(x-cos x))=

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