Q.14
The limit of the function
e-2t sin(t) as t → ∞ is _____
Why the Limit is 0
The function combines an exponentially decaying term e-2t, which approaches 0 rapidly as t grows, with the bounded oscillation of sin(t) between -1 and 1. The product e-2tsin(t) is squeezed between −e-2t and e-2t, both converging to 0. By the squeeze theorem, the limit equals 0.
Common Options Explained
Multiple-choice questions often test this limit with these choices:
| Option | Value | Explanation |
|---|---|---|
| A | 0 | Correct. Exponential decay e-2t→0 overrides bounded |sin(t)| ≤ 1 |
| B | 1 | Incorrect. No constant term exists; decay prevents approaching 1. |
| C | ∞ or -∞ | Incorrect. Exponential ensures values shrink toward 0, not diverge. |
| D | Does Not Exist (DNE) | Incorrect. Oscillation amplitude decreases to 0, unlike sin(t) alone. |
Mathematical Proof
Consider −e-2t ≤ e-2tsin(t) ≤ e-2t for all t≥0. Since limt→∞−e-2t=0 and limt→∞e-2t=0, the squeeze theorem confirms limt→∞e-2tsin(t)=0. This holds for any a>0 in e-atsin(bt).
The limit of e-2tsin(t) as t→∞ is 0.
