 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2007/paper-1/section-a/10/b/solution

The denominator vanishes only at

x = 0

, giving a vertical asymptote. Since

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

, the function tends to

+ \infty

from the left and

- \infty

from the right. Its derivative is

\frac{d}{d x} \frac{1}{1 - e ^{x}} = \frac{e ^{x}}{( 1 - e ^{x} ) ^{2}} > 0,

so each branch is increasing. The horizontal asymptotes are

y = 1

x \to - \infty

and

y = 0

x \to + \infty

. The left branch lies above

1

; the right branch lies below

0

, as in the preceding figure.

 New to topics? Read the docs here!