Question

Leila must also determine hunting policies to sustain a

population $W_0$of wolves that satisfy federal guidelines,

while maximizing the sustained elk population $E_0$for

which the state can sell hunting tags. Her predator-prey

models approximate the number of elk over time as

localid="1648495109241" $E\left(t\right)=\frac{E_0+72e^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)+8t{e}^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)}{1+0·2W_0}$

$\left(a\right)$Use the Squeeze Theorem for Limits to show that the

population goes toward localid="1648493738519" $\frac{E_0}{1+0·2W_0}ast\to \infty$

$\left(b\right)$Explain why L’Hopital’s Rule is ˆ not a good method for

calculating this limit.

Short answer

Part (a) Therefore the population goes towards $\frac{E_0}{1+0·2W_0}ast\to \infty$

Part (b) The L’Hopital’s Rule is not good for calculating limits

Step-by-step solution

Part(a) Step 1. Given information 

Given$E\left(t\right)=\frac{E_0+72e^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)+8t{e}^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)}{1+0·2W_0}$

Part (a) Step 2. To show the population goes towards E01+0·2W0 as t→∞

$$\begin{gathered} \underset{t\to \infty }{\lim }E\left(t\right)=\underset{t\to \infty }{\lim }\frac{E_0+72e^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)+8t{e}^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)}{1+0·2W_0} \\ \underset{t\to \infty }{\lim }E\left(t\right)=\underset{t\to \infty }{\lim }\frac{E_0}{1+0·2W_0}+\underset{t\to \infty }{\lim }\frac{72e^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)+8t{e}^{-0.006t}\sin \left(\mathrm{\pi t}/4\right)}{1+0·2W_0} \\ =\frac{E_0}{1+0·2W_0}+\frac{72\times 0\times \sin \left({\displaystyle \frac{\mathrm{\pi }\times 0}{4}}\right)+8\times \infty \times \sin \left({\displaystyle \frac{\mathrm{\pi }\times 0}{4}}\right)}{1+0·2W_0} \\ =\frac{E_0}{1+0·2W_0} \end{gathered}$$

therefore the population goes towards $\frac{E_0}{1+0·2W_0}ast\to \infty$

Part (b) Step 1. Explanation

here differentiate the terms and substitute the value t=0

The easiest way to find the limits by splitting the terms then apply the limits seperately.

Therefore the L’Hopital’s Rule is not good for calculating this limits.

The L’Hopital’s Rule is not good for calculating limits