 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015/2/b/solution

Combine the terms before taking their limit:

\frac{1}{x} - \frac{1}{e ^{x} - 1} = \frac{e ^{x} - 1 - x}{x ( e ^{x} - 1 )} .

x^{2} /2 + O (x^{3})

and the denominator is

x^{2} + O (x^{3})

. Consequently

x \to 0 lim (\frac{1}{x} - \frac{1}{e ^{x} - 1}) = \frac{1}{2} .

The cancellation removes the two individual singularities. Subtracting their separate divergent limits would not determine the finite answer.

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