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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/20/a/ii/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 20 / a / ii / Solution by

Codex  0 2026-10-18

At

- a

, the prefactor

(x - a) / x

tends to

2

, so the nonvanishing amplitude leaves the oscillatory discontinuity intact. At zero, use

sin (tan x) = x + O (x^{3})

to obtain

f_{1} (x) = (1 - \frac{a}{x}) [x + O (x^{3})] = - a + x + O (x^{2}) .

The limit is

- a

, whereas the assigned value is zero. This is a removable discontinuity, not a continuous crossing.

At

a

, write

x = a + h

. The estimate

∣ f_{1} (a + h) ∣ \leq \frac{∣ h ∣}{∣ a + h ∣} ⟶ 0

proves continuity at the assigned zero. But its difference quotient is

\frac{f _{1} ( a + h ) - f _{1} ( a )}{h} = \frac{s i n ( t a n ( a + h ))}{a + h} .

The sequences from part (i) make this approach

1/ a

and

- 1/ a

, so the derivative does not exist. This illustrates the distinct continuity and differentiability thresholds for a damped oscillation.

Therefore the discontinuities are $x = - π /2$ and $x = 0$ ; the nondifferentiability points are $x = - π /2, 0, π /2$ . The function is smooth elsewhere.

Figure 1.
Linear suppression of the positive-pole oscillations, with a removable discontinuity at zero
. Details.

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