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Limit

Evaluate the limit

    \[\ell =\lim_{x\rightarrow a} \frac{x^a-a^x}{x^x-a^a}\]

Solution

Rewrite the limit as

    \[\lim_{x\rightarrow a} \frac{x^a-a^x}{x^x-a^a} = \lim_{x\rightarrow a} \frac{\frac{x^a-a^a}{x-a} - \frac{a^x-a^a}{x-a}}{\frac{x^x-a^a}{x-a}}\]

Using the definition of the derivative we get that limit equals to

    \[\ell = \frac{1- \ln a}{1+\ln a}\]

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