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(Advance note: I'm not looking for the answers to this question. I want to understand what "the magnitude of the rate of change of $\theta$ increases without bound" means.)

Here's the problem I'm given: A boat is towed toward a dock by a cable attached to a winch that stands 14 ft above the water level. Let $\theta$ be the angle of the elevation of the winch and let L be the length of the cable as the boat is towed toward the dock.

Find $\lim_{L\to 14+} \frac{d\theta}{dL}$

Since I'm not including a drawing, let me make it clear that sin$\theta$ = $\frac{14}{L}$

I've worked through this problem and can get the correct answers. But, what in the world does it all mean?

Specifically, at the end, when I determine the limit is -$\infty$, this apparently means that, as the last foot of cable (L) is reeled in, the magnitude of the rate of change of $\theta$ increases without bound. What does this mean exactly in real world terms? Any insight is appreciated.

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It means that as you reel in the last bit of cable, you stop towing and start lifting vertically. Near that point, for a very small amount of extra cable wound, there is a large change in the magnitude of the angle of elevation (by magnitude, read absolute value).

Draw some diagrams with $L=50,51$ versus $L=15, 15.1$ and let $\Delta L=1$.

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