 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ia/paper-1/12e/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Changing variables gives

\frac{1}{h} \int_{a}^{x} (f (t + h) - f (t)) d t = \frac{1}{h} (\int_{x}^{x + h} f (s) d s - \int_{a}^{a + h} f (s) d s) .

[y, y + h]

tends to

f (y)

h \to 0

. The two terms therefore tend to

f (x)

and

f (a)

, proving the stated limit.

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