1. **Problem Statement:** Estimate the limit \(\lim_{t \to 0} \frac{e^t - 1}{t}\) using a table of values and graph. 2. **Formula and Important Rule:** The limit \(\lim_{t \to 0} \frac{e^t - 1}{t}\) is a classic limit that evaluates the derivative of \(e^t\) at \(t=0\). 3. **Table of Values:** | t (Left) | -0.1 | -0.01 | -0.001 | -0.0001 | -0.00001 | |----------|-------|--------|---------|----------|-----------| | f(t) = \frac{e^t - 1}{t} | \frac{e^{-0.1} - 1}{-0.1} \approx 0.9516 | \frac{e^{-0.01} - 1}{-0.01} \approx 0.9950 | \frac{e^{-0.001} - 1}{-0.001} \approx 0.9995 | \frac{e^{-0.0001} - 1}{-0.0001} \approx 0.99995 | \frac{e^{-0.00001} - 1}{-0.00001} \approx 0.999995 | | t (Right) | 0.00001 | 0.0001 | 0.001 | 0.01 | 0.1 | |-----------|---------|---------|-------|-------|-----| | f(t) = \frac{e^t - 1}{t} | \frac{e^{0.00001} - 1}{0.00001} \approx 1.000005 | \frac{e^{0.0001} - 1}{0.0001} \approx 1.00005 | \frac{e^{0.001} - 1}{0.001} \approx 1.0005 | \frac{e^{0.01} - 1}{0.01} \approx 1.005 | \frac{e^{0.1} - 1}{0.1} \approx 1.0517 | 4. **Intermediate Work:** As \(t\) approaches 0 from both sides, the values of \(\frac{e^t - 1}{t}\) approach 1. 5. **Graph Explanation:** The graph of \(y = \frac{e^t - 1}{t}\) near \(t=0\) shows the function approaching the value 1 smoothly from both sides. 6. **Conclusion:** Therefore, \[\lim_{t \to 0} \frac{e^t - 1}{t} = 1.\] This matches the derivative of \(e^t\) at 0, which is 1.