Let a(x) = N(0,1) and let b(x) = C(0,1). I would like to prove that the ratio of $$\frac{a(x)}{b(x)} \le \sqrt{\frac{2\pi}{e}}$$
Doing algebraic simplification yield something like $$(1 + x^2)e^{\frac{1}{2} - \frac{x^2}{2}} \le 2$$ I am stuck here and don't know how to proceed further.