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lim(x to 0) {(1+x)^((2)/(x))} (where {.}...

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

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

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

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  3. lim(x to 0) {(1+x)^((2)/(x))} (where {.} denotes the fractional part o...

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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 to 0) (sin(x^(2)))/("ln"(cos(2x^(2)-x))) is equal to

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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) (nx^(n-1)-(n+1)x^(n)+1)/((e^(x)-e)sinpix), where n=100,is eq...

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  10. lim(xto0) (log(1+x+x^(2))+log(1-x+x^(2)))/(secx-cosx)=

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  11. The value of lim(xto0) (root(3)(x^(3)+2x^(2))-sqrt(x^(2)+x)) is

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  12. The value of lim(xto0) (1+sinx-cosx+log(1-x))/(x^(3)) is

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  13. lim(xto0) (cos(tanx)-cosx)/(x^(4)) is equal to

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

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

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  16. lim(xto0) ((1+tanx)/(1+sinx))^(cosecx) is equal to

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  17. The value of lim(xto1) (2-x)^(tan((pix)/(2))) is

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  18. The value of lim(mtooo) ("cos"(x)/(m))^(m) is

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  19. lim(ntooo) ((n^(2)-n+1)/(n^(2)-n-1))^(n(n-1)) is equal to

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

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