31. (a) Prove that for all x > 0, 1 x 2 * 1 1 x2 , ex2/2 < 1 -(x) < 1

31.

(a) Prove that for all x > 0, 1 x √2π * 1 − 1 x2 , e−x2/2 < 1 − -(x) < 1 x √2π e−x2/2 . Hint: Integrate the following inequalities: (1 − 3y−4 )e−y2/2 < e−y2/2 < (1 + y−2 )e−y2/2 .

(b) Use part

(a) to prove that 1 − -(x) ∼ 1 x √2π e−x2/2 . That is, as x → ∞, the ratio of the two sides approaches 1.