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x x (x ) Towers of exponents The functions f(x) = (x ) and
Chapter 4, Problem 73RE(choose chapter or problem)
The functions \(f(x)=(x^x)^x\) and \(g(x)=x^{(x^{x})}\) are different functions. For example, f(3) = 19, 683 and \(g(3) \approx\ 7.6 \times 10^{12}\). Determine whether \(\lim _{x \rightarrow 0^+}\ f(x)\) and \(\lim _{x \rightarrow 0^+}\ g(x)\) are indeterminate forms and evaluate the limits.
Questions & Answers
QUESTION:
The functions \(f(x)=(x^x)^x\) and \(g(x)=x^{(x^{x})}\) are different functions. For example, f(3) = 19, 683 and \(g(3) \approx\ 7.6 \times 10^{12}\). Determine whether \(\lim _{x \rightarrow 0^+}\ f(x)\) and \(\lim _{x \rightarrow 0^+}\ g(x)\) are indeterminate forms and evaluate the limits.
ANSWER:Solution Step 1 x In this problem we are given with two functions namely f(x) = (x ) and g(x) = x x (x . We have to check whether lim f(x)and +im g(x)are in +ndeterminate forms and then x0 x0 we have to evaluate the limits. If they are in indeterminate form, we can evaluate the limit using l’hopital’s rule. l'Hôpital's Rule: f(x) 0 f(x) ± Suppose that we have one of the following cases, lim xa g(x) = 0r lixa g(x)= ± Where a can be any real number, infinity or negative infinity. f(x) f(x) In these cases we have lim g(x)= lim gx) xa xa