Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 19 / c

A trigonometric simplification avoids repeated differentiation:

\frac{t a n x - s i n x}{s i n ^{3} x} = \frac{1 - c o s x}{c o s x s i n ^{2} x} = \frac{1}{c o s x ( 1 + c o s x )}

for sufficiently small nonzero

x

. Continuity of cosine then gives

x \to 0 lim \frac{tan x - sin x}{sin ^{3} x} = \frac{1}{2} .

ii

This is the defining difference quotient for the derivative of sine at

a

. Therefore

x \to a lim \frac{sin x - sin a}{x - a} = cos a .

The result holds for every real constant

a

For every fixed real

a

, the base is positive once

x > ∣ a ∣

. Taking its logarithm gives

lo g [(\frac{x + a}{x - a})^{x}] = x [lo g (1 + \frac{a}{x}) - lo g (1 - \frac{a}{x})] .

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

, so this logarithm tends to

2 a

. Continuity of the exponential gives

x \to \infty lim (\frac{x + a}{x - a})^{x} = e^{2 a} .

For

a = 0

the base is identically one, consistently giving the same answer.

There are no discussions about this article yet.

There are currently no matching articles.