Chapter 8.7, Problem 111E

(a)

To determine

To prove: That the indeterminate form $0^0,{\infty }^0,\text{and}{1}^{\infty }$ do not always have a value of 1 by evaluating the limit $\underset{x\to 0}{\lim }{x}^{\left(\ln 2\right)/\left(1+\ln x\right)}$.

(b)

To determine

To prove: That the indeterminate form $0^0,{\infty }^0,\text{and}{1}^{\infty }$ do not always have a value of 1 by evaluating the limit $\underset{x\to \infty }{\lim }{x}^{\ln 2/1+\ln x}$.

(c)

To determine

To prove: That the indeterminate form $0^0,{\infty }^0,\text{and}{1}^{\infty }$ do not always have a value of 1 by evaluating the limit $\underset{x\to 0}{\lim }{\left(x+1\right)}^{\left(\ln 2\right)/\left(x\right)}$.