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The value of lim(x->0)((1+2x)/(1+3x))^(1...

The value of `lim_(x->0)((1+2x)/(1+3x))^(1/x^2)e^(1/x)` is `e^(5/2)` b. `e^2` c. `e^(-2)` d. 1

A

`e((5)/(2))`

B

`e^(2)`

C

`e^(-2)`

D

1

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(xrarr0)(lim)e^((1)/(x^(2))log(1+2x)-log(1+3x)+(1)/(x))`
`=e^(underset(xrarr0)(lim)(log(1+2x)-log(1+3x)+x)/(x^(2)))`
`=e^(underset(xrarr0)(lim)(2x-((2x)^(2))/(2)-3x+((3x)^(2))/(2)+x)/(x^(2)))" (using expansion)"`
`=e^(underset(xrarr0)(lim)((5)/(2)x^(2))/(x^(2)))`
`=e^((5)/(2))`
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