Limits with variable substitution and Trig \(\displaystyle\lim_{{{x}\to{0}}}{\frac{{{{\sin}^{{2}}{\left({3}{x}\right)}}}}{{{x}^{{2}}{\cos{{x}}}}}}=\lim_{{{u}\to{0}}}{\frac{{{{\sin}^{{2}}{\left({u}\right)}}}}{{{\left({\frac{{u}}{{3}}}\right)}^{{2}}}}}={9}{\left(\lim_{{{u}\to{0}}}{\frac{{{\sin{{u}}}}}{{u}}}\right)}^{{2}}\)

Question

Limits with variable substitution and Trig

lim_{x \to 0} \frac{\sin^{2} (3 x)}{x^{2} \cos x} = lim_{u \to 0} \frac{\sin^{2} (u)}{{(\frac{u}{3})}^{2}} = 9 {(lim_{u \to 0} \frac{\sin u}{u})}^{2}

Answer 1

Hint:

lim_{x \to 0} \frac{\sin^{2} (3 x)}{x^{2} \cos x} =

lim_{x \to 0} \frac{\sin (3 x)}{x} . lim_{x \to 0} \frac{\sin (3 x)}{x} . lim_{x \to 0} \frac{1}{\cos x} =

lim_{x \to 0} 3 \frac{\sin (3 x)}{3 x} . lim_{x \to 0} 3 \frac{\sin (3 x)}{3 x} . lim_{x \to 0} \frac{1}{\cos x} =

9 \underset{1}{\underset{⏟}{lim_{x \to 0} \frac{\sin (3 x)}{3 x}}} . \underset{1}{\underset{⏟}{lim_{x \to 0} \frac{\sin (3 x)}{3 x}}} . lim_{x \to 0} \frac{1}{\cos x} =

9.1 .1 . lim_{x \to 0} \frac{1}{\cos x} = 9.1 .1 .1

Answers (1)