Evaluate`lim_(xto0^(+)) x^(x)"and "lim_(xto0^(+))x^(x^(x))`

Options

A. `1//2`
B. 2
C. 1
D. None of these

Answer

`1" and "0`

Explanation

Let`y=x^(x)`
`:.logy=xlogx`
`:.underset(yto0)limlogy=underset(xto0)limxlogx`
`=underset(xto0)lim(logx)/(1//x)" "((oo)/(oo)" form")`
`=underset(xto0)lim(1//x)/(-1//x^(2))`
`=underset(xto0)lim(logx)/(1//x)" "((oo)/(oo)" form")`
`=underset(xto0)lim(1//x)/(-1//x^(2))`
`:." "y=1`
`=underset(xto0^(+))limx^(x^(x))=(0^(+))^(1)=0`