GATE 2022 EC – Question 33
Let $x_1(t)=e^{-t}u(t)$ and $x_2(t)=u(t)-u(t-2)$, where $u(\cdot)$ denotes the unit step function.
If $y(t)$ denotes the convolution of $x_1(t)$ and $x_2(t)$, then $\lim_{t\to\infty}y(t)=$ ________ (rounded off to one decimal place).
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Correct answer: 0
Explanation
For $t>2$, $y(t)=\int_{t-2}^{t}e^{-\tau}\,d\tau=e^{-(t-2)}-e^{-t}=e^{-t}(e^2-1)$, which tends to $0$ as $t\to\infty$.