Chapter 3.6, Problem 58ES

There is a myth that circulates among beginning calculus students which states that all indeterminate forms of types $0^0,{\infty }^0$ , and $1^{\infty }$ have value $1$ because “anything to the zero power is 1� and “ $1$ to any power is $1$ .� The fallacy is that $0^0,{\infty }^0$ , and $1^{\infty }$ are not powers of numbers, but rather descriptions of limits. The following examples, which were suggested by Prof. Jack Staib of Drexel University, show that such indeterminate forms can have any positive real value:

(a) $\underset{x\to 0^{+}}{\lim }\left[x^{\left(\ln a\right)/\left(1+\ln x\right)}\right]=a\left(\text{form }{0}^0\right)$

(b) $\underset{x\to +\infty }{\lim }\left[x^{\left(\ln a\right)/\left(1+\ln x\right)}\right]=a\left(\text{form }{\infty }^0\right)$

(c) $\underset{x\to 0}{\lim }\left[{\left(x+1\right)}^{\left(\ln a\right)/x}\right]=a\left(\text{form }{1}^{\infty }\right)$ .

Verify these results.