If \(\lim\limits_{x \rightarrow a}\frac{(a^x-x^a)}{x^x-a^a}\) = -1 then, what is the value of a?(a) 1(b) 2(c) 3(d) 4The question was posed to me by my college director while I was bunking the class.Origin of the question is First Order Derivative in section Limits and Derivatives of Mathematics – Class 11

If \(\lim\limits_{x \rightarrow a}\frac{(a^x-x^a)}{x^x-a^a}\) = -1 the

1.	If \(\lim\limits_{x \rightarrow a}\frac{(a^x-x^a)}{x^x-a^a}\) = -1 then, what is the value of a?(a) 1(b) 2(c) 3(d) 4The question was posed to me by my college director while I was bunking the class.Origin of the question is First Order Derivative in section Limits and Derivatives of Mathematics – Class 11
Answer» Right choice is (a) 1 For explanation: Let, y = x^x Thus, LOG y = x log x Differentiating both SIDES with respect to x, we get, 1/y dy/dx = (x1/x) + log x =>dy/dx = y(1 + log x) Or, dx^x/dx = x^x(1 + log x) Using, L’Hospital’s rule, \(\lim\limits_{x \rightarrow a}\frac{(a^x-x^a)}{x^x-a^a}\) = \(\lim\limits_{x \rightarrow a}\frac{(a^xlog⁡a-x^{a-1})}{x^x(1+logx)}\) = \(\lim\limits_{x \rightarrow a}\frac{(a^a*log⁡a-a^{a})}{a^a(1+loga)}\) = (log a – 1)/(log a + 1) As per the QUESTION, (log a – 1)/(log a + 1) = -1 Or, (log a – 1) = -log a – 1 Or, 2 log a = 0 Or, log a = 0 So, a = 1

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