Part BCSIR NET December 2023dalembert-limit

Dalembert limit

Consider the Cauchy problem for the wave equation $u_{t} t - 4 u_{xx} = 0$ on $R \times (0, \infty)$ , with u(x, 0) = $e^{- 1/ x^{2}}$ for x ≠ 0 (and 0 at x = 0), and u_t(x, 0) = x $e^{- x^{2}}$ . Which one of the following is true?

A. $l i m_{t \to \infty}$ u(5, t) = 1✓
B. $l i m_{t \to \infty}$ u(5, t) = 2
C. $l i m_{t \to \infty}$ u(5, t) = 1/2
D. $l i m_{t \to \infty}$ u(5, t) = 0

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 limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

Drill statements like this

Related counterexample: All the classical equations satisfy a maximum principle

Dalembert limit

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