My problem is to evaluate the following limit: $$\lim_{x \to +0}\frac{1}{x}{\int^{2022x}_{0}{t\,\sqrt{|\cos(\frac{1}{t})|}\,dt}}$$
I have no idea where to begin.
2026-04-09 15:16:39.1775747799
Evaluating $\lim_{x \to +0}\frac{1}{x}{\int^{2022x}_{0}{t\,\sqrt{|\cos(\frac{1}{t})|} \,dt}}$ without L'Hopital's Rule
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HINT: the inequality $0 \le |\cos\big(\frac{1}{t}\big)| \le 1$ holds for all real positive $t$. Thus, for each real positive $x$, the following string of inequalities hold: $$0 \ \le \ \frac{1}{x}\times\left|\int^{2022x}_{0}{t\sqrt{\left|\cos\left(\frac{1}{t}\right)\right|}\ dt}\right| \ \le \ \frac{1}{x}\times\left|\int^{2022x}_0 t \ dt\right|$$ $$=\frac{1}{x}\times1011x^2 \ = \ 1011x.$$
Can you take it from here.