Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure

Question

Find the limit $\lim \int_{(0,1]} f_n \, d\lambda $ where $ \lambda $ is lebesgue measure and $ f_n(x)=\dfrac{|\cos(x^{-2})|}{x^{1-1/n}} $ for $ x\in (0,1] $ Is there lebesgue integrable function $g$ such that $ |f_n(x)|\le g(x) $ for all $ n \ge 1 $?