 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/19/d/ii/solution

The original PDF specifies

x \to a

f (x) = x^{x} = e^{x l n x}

at the positive point

a

. The chain rule gives

f^{'} (x) = x^{x} (ln x + 1)

, so

x \to a lim \frac{x ^{x} - a ^{a}}{x - a} = a^{a} (1 + ln a), a > 0.

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