Home
Class 12
MATHS
Let alpha, beta epsilonR such that lim(x...

Let `alpha, beta epsilonR` such that `lim_(xto0)(x^(2)sin(betax))/(alpha x-sinx)=1` Then
`6(alpha+beta)`

A

6

B

7

C

2

D

12

Text Solution

Verified by Experts

The correct Answer is:
B
We find that
`lim_(xto0)(x^2sinbetax)/(alphax-sinx)`
` =lim_(xto0) (x(sin betax))/(a-(sinx)/(x))=(0)/(alpha-1)=0, if alpha ne 1`.
But, it is given that `lim_(xto 0) (x^2sinbetax)/(alphax-sin)=1.So, alpha=1`
Now,
`lim_(xto0) (x^2sinbetax)/(alphax-sinx)=1`
`rArr lim_(xto0) (beta^3((sinbetax)/(betax))=1`
`rArr lim_(xto0) (((sinbeta x)/(beta x)))/(((x-sinx)/(x^3)))=(1)/(beta)`
`rArr 6=(1)/(beta) rArr beta =(1)/(beta)" " [because lim_(xto0) (x-sinx)/(x^3)=(1)/(6)]`
Hence, `6(alpha+beta) =6 (1+(1)/(6))=7`.
Promotional Banner

Topper's Solved these Questions

  • LIMITS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section I - Solved Mcqs|101 Videos

  • LIMITS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|5 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|30 Videos

  • MATHEMATICAL INDUCTION

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|30 Videos

Similar Questions

Explore conceptually related problems

Let alpha,betainR be such that lim_(xto0) (x^(2)sin(betax))/(alphax-sinx)=1 . Then 6(alpha+beta) equals___________.

Let alpha, beta in R such that lim_(x ->0) (x^2sin(betax))/(alphax-sinx)=1 . Then 6(alpha + beta) equals

If lim _(xto0) (ln cot ((pi)/(4) -beta x))/( tan ax)=1, then (alpha)/(beta)=.....

If cos(alpha+beta)=0 then sin(alpha+2beta)=

The value of lim_(x to 0) ("sin" alpha X - "sin" beta x)/(e^(alphax) - e^(beta x)) equals

Find the value of x such that ((x+alpha)^n-(x+beta)^n)/(alpha-beta)=(sin (ntheta))/(sin^n theta) , where alpha and beta are the roots of the equation t^2-2t+2=0 .

Let f(alpha, beta) =|(cos (alpha + beta), -sin (alpha + beta), cos 2beta),(sin alpha, cos alpha, sin beta),(-cos alpha, sin alpha, cos beta)| The value of l=int_(0)^(pi//2)e^(beta)(f(0,0)+f((pi)/(2),beta)+f((3pi)/(2),(pi)/(2)-beta))dbeta is

Let f(alpha, beta) = =|(cos (alpha + beta), -sin (alpha + beta), cos 2beta),(sin alpha, cos alpha, sin beta),(-cos alpha, sin alpha, cos beta)| if i =int_(-pi//2)^(pi//2) cos ^(2)beta(f(0,beta)+f(0,(pi)/(2)-beta))dbeta then i is

If A(alpha, beta)=[("cos" alpha,sin alpha,0),(-sin alpha,cos alpha,0),(0,0,e^(beta))] , then A(alpha, beta)^(-1) is equal to

Prove that (sin alpha cos beta + cos alpha sin beta) ^(2) + (cos alpha coa beta - sin alpha sin beta) ^(2) =1.

OBJECTIVE RD SHARMA ENGLISH-LIMITS-Chapter Test
  1. Let alpha, beta epsilonR such that lim(xto0)(x^(2)sin(betax))/(alpha x...

    Text Solution

    |

  2. Let f(x)={(x^(2),x epsilonZ),((d(x^(2)-4))/(2-x),x !inZ):} the set of ...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  5. If x(1)=3 and x(n+1)=sqrt(2+x(n))" ",nge1, then underset(ntooo)limx(n)...

    Text Solution

    |

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

    Text Solution

    |

  7. Evaluate underset(ntooo)limncos((pi)/(4n))sin((pi)/(4n)).

    Text Solution

    |

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

    Text Solution

    |

  9. If f(x) is the integral of (2 sin x - sin 2x )/(x ^ 3 ) , w...

    Text Solution

    |

  10. Evaluate: ("lim")(xvec0)x^m(logx)^n ,m , n in Ndot

    Text Solution

    |

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

    Text Solution

    |

  12. underset(xtoa)lim(log(x-a))/(log(e^(x)-e^(a)))

    Text Solution

    |

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

    Text Solution

    |

  14. If f(a)=2,f^(prime)(a)=1,g(a)=-1,g^(prime)(a)=2, then the value of ("l...

    Text Solution

    |

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

    Text Solution

    |

  16. A(i)=(x-a(i))/(|x-a(i)|),i=1,2,...,n," and "a(1)lta(2)lta(3)lt...lta(n...

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

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

    Text Solution

    |

  20. Evaluate the following limits : Lim(x to oo) sqrt(((x+sin x)/(x- cos...

    Text Solution

    |

  21. Evaluate: ("lim")(xvecoo)(1+1/(a+b x))^(c+dx),w h e r ea , b , c ,a n ...

    Text Solution

    |