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If ("lim")(xvec0)[1+x+(f(x))/x]^(1/x)=e^...

If `("lim")_(xvec0)[1+x+(f(x))/x]^(1/x)=e^3,` then find the value of `1n(("lim")_(xvec0)[1+(f(x))/x]^(1/x))i s____`

A

e

B

`e^(2)`

C

`e^(3)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(xrarr0)(lim)[1+x+(f(x))/(x)]^(1//x)=e^(3)`
`rArr" "underset(xrarr0)(lim)e^(underset(xrarr0)(lim)[1+x+(f(x))/(x^(2))-1](1)/(x))=e^(3)`
`rArr" "underset(xrarr0)(lim)e^(underset(xrarr0)(lim)[1+(f(x))/(x^(2))])=e^(3)`
`rArr" "underset(xrarr0)(lim)(f(x))/(x^(2))=2`
Now, `underset(xrarr0)(lim)[1+(f(x))/(x)]^(1//x)=e^(underset(xrarr0)(lim)[1+(f(x))/(x)-1](1)/(x))=e^(underset(xrarr0)(lim)(f(x))/(x^(2)))=e^(2)`
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CENGAGE-LIMITS-Single Correct Answer Type
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