 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/2/b/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The numerator and denominator both vanish at zero. The fundamental theorem of calculus differentiates the numerator, while the denominator derivative is nonzero near zero. L'Hôpital's rule therefore gives

x \to 0 lim \frac{\int _{0}^{x} e ^{- y^{2}} d y}{3 x - x ^{3}} = x \to 0 lim \frac{e ^{- x^{2}}}{3 - 3 x ^{2}} = \frac{1}{3} .

 New to topics? Read the docs here!