My problem says, "there's a myth circulating among calculus students which states that all indeterminate forms of types
and 1^(infinity) have a value of 1 since anything to the zero power is 1 and 1 to any power is 1.
But the thing is, those 3 things I mentioned before are "descriptions of limits" rather than powers of numbers. Please show that such indeterminate forms can have any positive real value.
(PS: Apologize for the terrible typing, I don't know how to use latex well.)
1)
lim (as x-> 0+)
[x^(lna/(1+lnx))] =
= a
2)
lim (as x-> 0+)
[x^(lna/(1+lnx))] = infinity^0 = a
3)
lim (as x-> 0+)
[(x+1)^(lna/x)] = 1^(infinity) = a
I am really stumped on how to do these. I did #1 and ended up with 1/(a(x+1)) which means that it would end up with a fraction of 1/a if x->0, which does not equal a.
My attempt
x^(lna/(1+lnx))
I used the e^ln thing to get the exponent down. Then I used L'Hopital rule.
e^(1/(a(x+1)))
?