Part CCSIR NET December 2023vanishing-factor-at-endpoint

Vanishing factor at endpoint

Let $(f_{n})$ be the sequence of functions on [0, 1] defined by $f_{n} (x) = x^{n} lo g ((1 + x) /2)$ . Which of the following statements are true?

A. $(f_{n})$ converges pointwise on [0, 1].✓
B. $(f_{n})$ converges uniformly on compact subsets of [0, 1) but not on [0, 1).
C. $(f_{n})$ converges uniformly on [0, 1) but not on [0, 1].
D. $(f_{n})$ converges uniformly on [0, 1].✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: fₙ → f uniformly ⇒ fₙ′ → f′

The chapter behind this: Uniform convergence — what it buys and how to test it — free to read

From Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini

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