limX^(1/x)
x is approaching infinity from both sides.
use l'hospital's rule
x is approaching infinity from both sides.
use l'hospital's rule
-
x^(1/x)
Take the natural logarithm to bring the exponent down:
ln(x^(1/x))
(1/x)(lnx)
(lnx)/x
Now that it's in a form we can deal with (infinity/infinity), take the derivative of the top and bottom individually:
(d/dx)[lnx] / (d/dx)[x]
(1/x) / 1
1/x
Now the limit as x->inf is 0, since the denominator gets bigger and bigger, the expression approaches 0.
But that's not the final answer. Originally we took the natural log, so to reverse that, we "exponentiate", or make our answer the power of e. So:
e^0 = 1
Hope that helps :)
Take the natural logarithm to bring the exponent down:
ln(x^(1/x))
(1/x)(lnx)
(lnx)/x
Now that it's in a form we can deal with (infinity/infinity), take the derivative of the top and bottom individually:
(d/dx)[lnx] / (d/dx)[x]
(1/x) / 1
1/x
Now the limit as x->inf is 0, since the denominator gets bigger and bigger, the expression approaches 0.
But that's not the final answer. Originally we took the natural log, so to reverse that, we "exponentiate", or make our answer the power of e. So:
e^0 = 1
Hope that helps :)