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Question

# Numerical and Graphical Reasoning A crossed belt connects a 20-centimeter pulley (10-cm radius) on an electric motor with a 40 -centimeter pulley (20-cm radius) on a saw arbor (see figure). The electric motor runs at 1700 revolutions per minute.Find$\lim _{\phi \rightarrow(\pi / 2)^{-}} L .$Use a geometric argument as the basis of a second method of finding this limit.

Solution

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In part $(c)$ of this task we have found the function $L$ that describes the length of the belt in relation to the angle $\phi$ as shown in the picture. That function is:

$L\left(\phi\right)=30\left(\pi+2\phi\right)+\frac{60}{\tan{\phi}}\text{.}$

We get:

\begin{align*} \lim_{\phi\to\frac{\pi}{2}^{-}}{L}&=\lim_{\phi\to\frac{\pi}{2}^{-}}{\left[30\left(\pi+2\phi\right)+\frac{60}{\tan{\phi}}\right]}\\ &=\lim_{\phi\to\frac{\pi}{2}^{-}}{\left[30\left(\pi+2\phi\right)\right]}+\lim_{\phi\to\frac{\pi}{2}^{-}}{\left[\frac{60}{\tan{\phi}}\right]}\\ &=30\cdot\left(\pi+2\cdot\frac{\pi}{2}\right)+0\tag{1}\\ &=30\cdot2\pi\\ &=60\pi\text{.} \end{align*}

In step $(1)$ we get that the second limit is $0$. The second limit is zero because the tangent function is not defined at $x=\frac{\pi}{2}$. When $x$ approaches $\frac{\pi}{2}$ from the left side, the tangent function approaches positive infinity, and thus $\frac{60}{\tan{\phi}}$ approaches $0$.

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