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The value of lim(xtooo) ((2^(x^(n)))e^((...

The value of `lim_(xtooo) ((2^(x^(n)))e^((1)/(x))-(3^(x^(n)))e^((1)/(x)))/(x^(n))` (where `n in N`) is

A

`e`

B

`e^(2)`

C

`e^(-1)`

D

1

Text Solution

Verified by Experts

The correct Answer is:
B
`L=underset(xtooo)lim((2^(x^(n)))e^((1)/(x))-(3^(x^(n)))e^((1)/(x)))/(x^(n))=underset(xtooo)lim((3)^((x^(n))/(e^(x)))(((2)/(3))^((x^(n))/(e^(x)))-1))/(x^(n))`
Now, `underset(xtooo)lim(x^(n))/(e^(x))=underset(xtooo)lim(n!)/(e^(x))=0`
(Differentiating numerator and denominator `n` times for L'Hospital's rule)
Hence, `L=underset(xtooo)lim(3)^((x^(n))/(e^(x)))underset(xtooo)lim((((2)/(3))^((x^(n))/(e^(x)))-1))/((x^(n))/(e^(x)))underset(xtooo)lim(1)/(e^(x))`
`=1xxlog(2//3)xx0=0`
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CENGAGE-LIMITS-Exercise (Single)
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  11. The value of lim(xto0) (1+sinx-cosx+log(1-x))/(x^(3)) is

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

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

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

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