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Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fold convolution. Let $f(t)=\lim _{n \rightarrow \infty} f_{n}(t)$. Then, which of the following is TRUE?

  1. The area under the curve of $f(t)$ is zero.
  2. The area under the curve of $f(t)$ is $\infty$.
  3. $f(t)$ has width $\infty$ and height $1$ .
  4. $f(t)$ has width $0$ and height $\infty$.
  5. None of the above.
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