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Evaluate lim(xto0) (x(e^(x)-1))/(1-cosx)...

Evaluate `lim_(xto0) (x(e^(x)-1))/(1-cosx)` is equal to

A

`e`

B

`(1)/(e)`

C

1

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
D
`underset(xto0)lim(x(e^(x)-1))/(1-cosx)=underset(xto0)lim(2x(e^(x)-1))/(4"sin"^(2)(x)/(2))`
`=2underset(xto0)lim[((x//2)^(3))/("sin"^(2)(x)/(2))]((e^(x)-1)/(x))=2`
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CENGAGE PUBLICATION-LIMITS-Exercises (Single Correct Answer Type)
  1. lim(x to pi/2)sin(xcosx)/("cos"(xsinx)) is equal to (a) 0 (b) pi/2 ...

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  2. lim(xto0) [(1-e^(x))(sinx)/(|x|)] is (where [.] represents the greates...

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  3. Evaluate lim(xto0) (x(e^(x)-1))/(1-cosx) is equal to

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  4. If f(x)=lim(ntooo) n(x^(1//n)-1)," then for "xgt0, ygt0,f(xy) is equal...

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  5. lim(x->0) {(1+x)^(2/x)} (where {.} denotes the fractional part of x ...

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  6. The value of lim(xtooo) ((2^(x^(n)))e^((1)/(x))-(3^(x^(n)))e^((1)/(x))...

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  7. ("lim")(x->0)(sin(x^2))/(1n(cos(2x^2-x))) is equal to (a) 2 (b) -2 ...

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  8. lim(xtooo) (e^(1//x^(2))-1)/(2tan^(-1)(x^(2))-pi) is equal to

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  9. lim(xto0) ((2^(m)+x)^(1//m)-(2^(n)+x)^(1//n))/(x) is equal to

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  10. The value of lim(ntooo) [(1)/(n)+(e^(1//n))/(n)+(e^(2//n))/(n)+...+(e^...

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  11. ("lim")(xto1)(n x^(n+1)-(n+1)x^n+1)/((e^x-e)sinpix),\ where \n=100 , ...

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  12. lim(x->0)(log(1+x+x^2)+"log"(1-x+x^2))/(secx-cosx)=

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  13. The value of lim(xtooo) (root(3)(x^(3)+2x^(2))-sqrt(x^(2)+x)) is

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  14. The value of lim(x->0)(1+sinx-cosx+"log"(1-x))/(x^3) is (a)1/2 ...

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  15. If lim(xtoa)f(x)=1 and lim(xtoa)g(x)=oo then lim(xtoa){f(x)}^(g(x))=e^...

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  16. If lim xto0(x^(-3)sin3x+a x^(-2)+b) exists and is equal to 0, then

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  17. If lim(xto0)(x^n-sinx^n)/(x-sin^n x) is non-zero finite, then n must b...

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  18. ("lim")(x -> 0)((1+tanx)/(1+sinx))^(cos e cx) is equal to (a)e ...

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  19. The value of lim(x->1)(2-x)^(tan((pix)/2)) is (a)e^(-2/pi) (b) e^...

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  20. The value of ("lim")(xvecoo)(cos"x"/"m")^("m") is 1 (b) e (c) e^(-1...

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