Exercises: Evaluate the following limits by constructing the table of values. ( 5 points each item) (Show step by step process on your answer sheet) 1. \( \lim _{x \rightarrow 2} 5^{x} \) 2. \( \lim _{x \rightarrow 4} \log x \) 3. \( \lim _{x \rightarrow \pi} \cos x \)

Sure, let's evaluate each limit by constructing tables of values as \( x \) approaches the target value. This method helps visualize the behavior of the function near the point of interest. --- ### 1. \( \lim_{x \rightarrow 2} 5^{x} \) **Step-by-Step Process:** 1. **Choose values of \( x \) approaching 2 from both sides:** - Left of 2: 1.9, 1.99, 1.999 - Right of 2: 2.1, 2.01, 2.001 2. **Compute \( 5^{x} \) for each chosen \( x \):** | \( x \) | \( 5^{x} \) | |---------|----------------------| | 1.9 | \( 5^{1.9} \approx 23.69 \) | | 1.99 | \( 5^{1.99} \approx 24.90 \) | | 1.999 | \( 5^{1.999} \approx 24.998 \) | | **2** | \( 5^{2} = 25 \) | | 2.001 | \( 5^{2.001} \approx 25.002 \) | | 2.01 | \( 5^{2.01} \approx 25.10 \) | | 2.1 | \( 5^{2.1} \approx 25.56 \) | 3. **Analyze the table:** - As \( x \) approaches 2 from both sides, \( 5^{x} \) approaches 25. **Conclusion:** \[ \lim_{x \rightarrow 2} 5^{x} = 25 \] --- ### 2. \( \lim_{x \rightarrow 4} \log x \) *(Assuming \( \log x \) refers to the natural logarithm, \( \ln x \))* **Step-by-Step Process:** 1. **Choose values of \( x \) approaching 4 from both sides:** - Left of 4: 3.9, 3.99, 3.999 - Right of 4: 4.001, 4.01, 4.1 2. **Compute \( \ln x \) for each chosen \( x \):** | \( x \) | \( \ln x \) | |---------|-----------------------| | 3.9 | \( \ln(3.9) \approx 1.360 \) | | 3.99 | \( \ln(3.99) \approx 1.384 \) | | 3.999 | \( \ln(3.999) \approx 1.386 \) | | **4** | \( \ln(4) \approx 1.386 \) | | 4.001 | \( \ln(4.001) \approx 1.386 \) | | 4.01 | \( \ln(4.01) \approx 1.390 \) | | 4.1 | \( \ln(4.1) \approx 1.414 \) | 3. **Analyze the table:** - As \( x \) approaches 4 from both sides, \( \ln x \) approaches approximately 1.386. **Conclusion:** \[ \lim_{x \rightarrow 4} \ln x = \ln 4 \approx 1.386 \] --- ### 3. \( \lim_{x \rightarrow \pi} \cos x \) **Step-by-Step Process:** 1. **Choose values of \( x \) approaching \( \pi \) from both sides:** - Left of \( \pi \): \( \pi - 0.1 \), \( \pi - 0.01 \), \( \pi - 0.001 \) - Right of \( \pi \): \( \pi + 0.001 \), \( \pi + 0.01 \), \( \pi + 0.1 \) 2. **Compute \( \cos x \) for each chosen \( x \):** Let's approximate \( \pi \approx 3.1416 \). | \( x \) | \( \cos x \) | |------------------|--------------------------| | \( \pi - 0.1 \approx 3.0416 \) | \( \cos(3.0416) \approx -0.995 \) | | \( \pi - 0.01 \approx 3.1316 \) | \( \cos(3.1316) \approx -0.99995 \) | | \( \pi - 0.001 \approx 3.1406 \) | \( \cos(3.1406) \approx -0.9999995 \) | | **\( \pi \approx 3.1416 \)** | \( \cos(\pi) = -1 \) | | \( \pi + 0.001 \approx 3.1426 \) | \( \cos(3.1426) \approx -0.9999995 \) | | \( \pi + 0.01 \approx 3.1516 \) | \( \cos(3.1516) \approx -0.99995 \) | | \( \pi + 0.1 \approx 3.2416 \) | \( \cos(3.2416) \approx -0.995 \) | 3. **Analyze the table:** - As \( x \) approaches \( \pi \) from both sides, \( \cos x \) approaches -1. **Conclusion:** \[ \lim_{x \rightarrow \pi} \cos x = -1 \] --- These step-by-step evaluations using tables of values demonstrate how each function behaves as \( x \) approaches the specified limit points.