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lim(x->0) {(1+x)^(2/x)} (where {.} denot...

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

(a) `e^2−7`

(b) `e^2−8`

(c) `e^2−6`

(d) none of these

A

1

B

`e`

C

`e^(-1)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(1+x)^(2//x)=(1+x)^(2//x)-[(1+x)^(2//x)]`
Now, `underset(xto0)lim(1+x)^(2//x)=e^(2)`
or`underset(xto0)lim{(1+x)^(2//x)}=e^(2)-[e^(2)]=e^(2)-7`
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CENGAGE PUBLICATION-LIMITS-Exercises (Single Correct Answer Type)
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  3. lim(x->0) {(1+x)^(2/x)} (where {.} denotes the fractional part of x ...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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  19. ("lim")(xvecoo)((n^2)/(n^2))^(n(n-1)i se q u a lto e (b) e^2 (c) e^(...

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  20. lim(ntooo) {((n)/(n+1))^(alpha)+"sin"(1)/(n)}^(n) (where alphainQ) is ...

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