# Limits with variable substitution and Trig \lim_{x\to0}\frac{\sin^2(3x)}{x^2\cos x}=\lim_{u\to0}\frac{\sin^2(u)}{(\frac u3)^2}=9(\lim_{u\to0}\frac{\sin u}u)^2

Limits with variable substitution and Trig
$\underset{x\to 0}{lim}\frac{{\mathrm{sin}}^{2}\left(3x\right)}{{x}^{2}\mathrm{cos}x}=\underset{u\to 0}{lim}\frac{{\mathrm{sin}}^{2}\left(u\right)}{{\left(\frac{u}{3}\right)}^{2}}=9{\left(\underset{u\to 0}{lim}\frac{\mathrm{sin}u}{u}\right)}^{2}$
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Hint:
$\underset{x\to 0}{lim}\frac{{\mathrm{sin}}^{2}\left(3x\right)}{{x}^{2}\mathrm{cos}x}=$
$\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(3x\right)}{x}.\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(3x\right)}{x}.\underset{x\to 0}{lim}\frac{1}{\mathrm{cos}x}=$
$\underset{x\to 0}{lim}3\frac{\mathrm{sin}\left(3x\right)}{3x}.\underset{x\to 0}{lim}3\frac{\mathrm{sin}\left(3x\right)}{3x}.\underset{x\to 0}{lim}\frac{1}{\mathrm{cos}x}=$
$9\underset{1}{\underset{⏟}{\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(3x\right)}{3x}}}.\underset{1}{\underset{⏟}{\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(3x\right)}{3x}}}.\underset{x\to 0}{lim}\frac{1}{\mathrm{cos}x}=$
$9.1.1.\underset{x\to 0}{lim}\frac{1}{\mathrm{cos}x}=9.1.1.1$