---
title: "Final value theorem for Z-transform"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:final-value-theorem-for-z-transform"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/final-value-theorem-for-z-transform
site: BCT Engineering
---

# Final value theorem for Z-transform

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Kartik (new course), 2 marks
- Video: [Z-Transforms-Final Value Theorem-Proof](https://www.youtube.com/watch?v=Ge3QVtqUCsw) — Btech Maths Hub
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/final-value-theorem-for-z-transform
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find x(∞) using the final value theorem of z-transform when x(t) = 1 − e^(−at).

## Short answer

X(z)=z/(z-1)-z/(z-e^{-aT}). (z-1)X(z) has pole inside unit circle, so final-value theorem applies. Limit gives x(∞)=1.

## Answer

**Final value theorem:** if X(z) = Z[x(k)] and (z − 1)X(z) has no poles on or outside the unit circle, then x(∞) = lim_{z→1} (z − 1) X(z).

Take x(kT) = 1 − e^(−akT), a > 0, so e^(−aT) < 1. Then

```text
X(z) = z/(z − 1) − z/(z − e^(−aT))
(z − 1) X(z) = z − z(z − 1)/(z − e^(−aT))
```

The only pole of (z − 1)X(z) is at z = e^(−aT), which lies inside the unit circle, so the theorem applies. Putting z = 1:

```text
x(∞) = 1 − 1·0/(1 − e^(−aT)) = 1
```

**Answer: x(∞) = 1** (agrees with 1 − e^(−a·∞) = 1).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The final value theorem states x(infinity) =
   A. lim as z to 1 of (z-1)X(z)
   B. lim as z to infinity X(z)
   C. lim as z to 1 of X(z)
   D. lim as z to 0 of (z-1)X(z)
   Answer: A. Provided (z-1)X(z) has no poles on or outside the unit circle. (Key point: Theorem)

2. Why can the theorem be used here?
   A. X(z) has no poles
   B. The only pole of (z-1)X(z) is e^(-aT), inside the unit circle
   C. a is negative
   D. T = 0
   Answer: B. a > 0 so e^(-aT) < 1. (Key point: Condition)

3. At z = 1, the term z(z-1)/(z - e^(-aT)) is
   A. 1
   B. infinity
   C. 0
   D. e^(-aT)
   Answer: C. The factor (z-1) vanishes while the denominator is nonzero. (Key point: Evaluation)

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