The z-transform of x[n] = a^n u[n] is X(z) = frac z z-a with ROC |z|>|a|. For a = 0.8, the final value lim_ n to infty x[n] is:

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Electrical Engineering QUESTION #2167

Question 1

The z-transform of $x[n] = a^n u[n]$ is $X(z) = \frac{z}{z-a}$ with ROC $|z|>|a|$. For $a = 0.8$, the final value $\lim_{n \to \infty} x[n]$ is:

$0$✔️
$1$
$0.8$
Diverges to infinity

Correct Answer Explanation

Since $|a| = 0.8 < 1$, $\lim_{n\to\infty}(0.8)^n = 0$. Confirmed by the Final Value Theorem: $\lim_{z\to1}(z-1)X(z) = \lim_{z\to1}\frac{z(z-1)}{z-0.8} = 0$.

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