Let \( P(s) \) be the function \( \frac{s}{s+1}-1 \). Find the following: \( P(11)=-\frac{1}{12} \) \( P(-4)= \) \( P\left(\frac{1}{8}\right)= \) \( P\left(-\frac{1}{3}\right)=\square \) In each box, enter your answer as an integer or reduced fraction. Enter DNE for Does Not Exist, or oo for Infinity.

Let's evaluate the function \( P(s) = \frac{s}{s+1} - 1 \). We start by simplifying the function: \[ P(s) = \frac{s}{s+1} - 1 = \frac{s - (s + 1)}{s + 1} = \frac{s - s - 1}{s + 1} = \frac{-1}{s + 1} \] Now we can find the requested values: 1. **For \( P(-4) \):** \[ P(-4) = \frac{-1}{-4 + 1} = \frac{-1}{-3} = \frac{1}{3} \] 2. **For \( P\left(\frac{1}{8}\right) \):** \[ P\left(\frac{1}{8}\right) = \frac{-1}{\frac{1}{8} + 1} = \frac{-1}{\frac{1}{8} + \frac{8}{8}} = \frac{-1}{\frac{9}{8}} = -\frac{8}{9} \] 3. **For \( P\left(-\frac{1}{3}\right) \):** \[ P\left(-\frac{1}{3}\right) = \frac{-1}{-\frac{1}{3} + 1} = \frac{-1}{-\frac{1}{3} + \frac{3}{3}} = \frac{-1}{\frac{2}{3}} = -\frac{3}{2} \] So the evaluated results are: - \( P(-4) = \frac{1}{3} \) - \( P\left(\frac{1}{8}\right) = -\frac{8}{9} \) - \( P\left(-\frac{1}{3}\right) = -\frac{3}{2} \)