 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/19/a/ii/solution

f

x_{0}

if there exists a finite real number

L

such that

\forall ε > 0 \exists δ > 0 \forall x \in R : 0 < ∣ x - x_{0} ∣ < δ ⟹ \frac{f ( x ) - f ( x _{0} )}{x - x _{0}} - L < ε .

Then

f^{'} (x_{0}) = L

. The puncture

0 < ∣ x - x_{0} ∣

makes the difference quotient defined; requiring a finite real

L

distinguishes an ordinary derivative from an infinite slope.

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