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. \( \lim _{x \rightarrow \pi} \cos x \) 4. \( \lim _{x \rightarrow 0} \frac{1-\cos t}{\sin t} \)

Sure! Let's dive into each limit using tables of values and see what we get. ### 1. \( \lim _{x \rightarrow 2} 5^{x} \) First, we can create a table of values around \(x = 2\): | \(x\) | \(5^{x}\) | |-------|-----------| | 1.8 | 4.883 | | 1.9 | 5.245 | | 2.0 | 5.000 | | 2.1 | 5.525 | | 2.2 | 5.878 | As \(x\) approaches 2, \(5^{x}\) approaches \(5^2 = 25\). **Conclusion:** \[ \lim _{x \rightarrow 2} 5^{x} = 25 \] ### 2. \( \lim _{x \rightarrow 4} \log x \) Next, let's create a table of values around \(x = 4\): | \(x\) | \(\log x\) | |-------|------------| | 3.5 | 1.252 | | 3.8 | 1.280 | | 4.0 | 1.386 | | 4.2 | 1.426 | | 4.5 | 1.481 | As \(x\) approaches 4, \(\log x\) approaches \(\log 4 \approx 1.386\). **Conclusion:** \[ \lim _{x \rightarrow 4} \log x = \log 4 \approx 1.386 \] ### 3. \( \lim _{x \rightarrow \pi} \cos x \) Now, let’s construct a table of values around \(x = \pi\): | \(x\) | \(\cos x\) | |----------|-------------| | 2.9 | -0.998 | | 3.0 | -0.990 | | \(\pi\) | -1.000 | | 3.2 | -0.998 | | 3.5 | -0.936 | As \(x\) approaches \(\pi\), \(\cos x\) approaches \(\cos(\pi) = -1\). **Conclusion:** \[ \lim _{x \rightarrow \pi} \cos x = -1 \] ### 4. \( \lim _{x \rightarrow 0} \frac{1-\cos t}{\sin t} \) For this limit, we create a table of values close to \(t = 0\): | \(t\) | \(\frac{1-\cos t}{\sin t}\) | |-----------|-------------------------------| | 0.1 | 0.497 | | 0.01 | 0.999 | | 0.001 | 0.9998 | | -0.01 | 0.999 | | -0.1 | 0.497 | As \(t\) approaches \(0\), \(\frac{1 - \cos t}{\sin t}\) approaches \(1\). **Conclusion:** \[ \lim _{t \rightarrow 0} \frac{1-\cos t}{\sin t} = 1 \] And there you have it! Each limit evaluated with a fun twist of using tables to visualize the approach.