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If lim(xrarr0)(ae^x-b cos x + ce^(-x))/(...

If `lim_(xrarr0)(ae^x-b cos x + ce^(-x))/(xsinx )=2`,then `a+b+c=`

A

2

B

4

C

0

D

6

Text Solution

Verified by Experts

The correct Answer is:
B
We observe that as `xto0`, numerator tends to `a-b+c` whereas the denominator tends to 0. therefore, for the limit to exist we must have.
`a-b+c=0` .......(i)
Now, if ` a-b+c=0`, then,
`lim_(xto0) (ae^x-b cos x +ce^-x)/(x sin x) ` is in the the `(0)/(0)` form.
`therefore lim_(xto0) (ae^(x)+bsinx-ce^(-x))/(sin x +cos x ) =2 ["Using L' Hospital's rule"]`.
Here, the numerator is tendor to `a-c as xto`. Therefore for the limit to exist, we must have
`a-c=0` .......(ii)
Now,if `a-c, =0`, then
`lim_(xto0) (ae^x+bsinx-ce^-x)/(sinx +x cos x)` is in the form `(0)/(0)`.
`therefore lim(xto0) (ae^x+bcos x +ce^-x)/(2cos x-x sin x) =2`
`rArr (a+b+c)/(2) =2rArr a+b+c=4`
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OBJECTIVE RD SHARMA ENGLISH-LIMITS-Chapter Test
  1. If lim(xrarr0)(ae^x-b cos x + ce^(-x))/(xsinx )=2,then a+b+c=

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

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

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

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

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  8. Evaluate ("lim")(n→oo){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 2x )/(x ^ 3 ) , w...

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

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

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

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

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

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

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

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

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

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

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