Find lim without L'Hopital rule and series expansions.
Could someone help me , I did not understand how to find it, thanks.
Answer
Recall \arctan y = \int_0^y 1/(1+t^2)\, dt. Verify that
1-t^2 \le \frac{1}{1+t^2}\le 1 -t^2 + t^4
for all t. Thus for y>0,
y-y^3/3 \le \arctan y \le y - y^3/3 + y^5/5.
It follow that our expression is bounded below by
\frac{p(\sqrt x/p - x^{3/2}/(3p^3)) - q(\sqrt x/q - x^{3/2}/(3q^3)+x^{5/2}/(5q^5))}{x^{3/2}}.
Simplify to see this \to 1/(3q^2) - 1/(3p^2). There is a similar estimate from above, giving the same limit. By the squeeze theorem, the limit is 1/(3q^2) - 1/(3p^2).
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