 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-2/12/a/iii/solution

e^{- x} = 1 - x + O (x^{2})

. Hence

1 - e^{- x} = x + O (x^{2})

, and

x \to 0 lim \frac{x}{1 - e ^{- x}} = 1.

The denominator is nonzero for every real

x \neq = 0

, so this is a two-sided finite limit.

 New to topics? Read the docs here!