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

Write the product as a quotient with indeterminate form $(-\infty )/(+\infty )$. By L'Hôpital's rule,

The original PDF specifies $x\to a$. The quotient is the difference quotient for the function $f(x)=x^x=e^{x\ln x}$ at the positive point $a$. The chain rule gives $f^{\prime }(x)=x^x(\ln x+1)$, so

Use the root limit of a finite sum of powers. Put $M=\max \{1,a,a^2/2\}\ge 1$. For every $x>0$, at least one of the three terms is $M^x$ and none is larger, soTaking the positive $x$th root givesSince $3^{1/x}\to 1$, the squeeze theorem givesIndeed, for $a\le 1$ both remaining bases are at most $1$; for $1\le a\le 2$, $a\ge a^2/2$; and for $a\ge 2$, $a^2/2\ge a$. The endpoint cases $a=0,1,2$ are included in the same bound.