Let's find the limit step by step: We are given: \[ \lim_{x \to 6^+} \frac{6}{x-6} \] ### **Step 1: Analyze the function** - The denominator \(x - 6\) approaches 0 as \(x\) approaches 6. - The numerator is a constant 6. - We are taking the **right-hand limit** (\(x \to 6^+\)), which means \(x\) is a little greater than 6. ### **Step 2: Substitute values approaching 6 from the right** Let's try some values a bit larger than 6: - For \(x = 6.1\): \(\frac{6}{6.1-6} = \frac{6}{0.1} = 60\) - For \(x = 6.01\): \(\frac{6}{6.01-6} = \frac{6}{0.01} = 600\) - For \(x = 6.001\): \(\frac{6}{6.001-6} = \frac{6}{0.001} = 6000\) As \(x\) gets closer and closer to 6 from the right, the denominator gets closer and closer to 0 but stays **positive**. The quotient grows **larger and larger positively**. ### **Step 3: Conclusion** Therefore, the limit is **positive infinity**. ### **Final Answer:** \[ \boxed{+\infty} \] **Explanation:** As \(x\) approaches 6 from the right, the denominator is a small positive number, and the whole fraction increases without bound. So the right-hand limit is \(+\infty\).

Let's analyze the limit \(\lim_{x \to 6^+} \frac{6}{x - 6}\) step by step, incorporating tables and graphs to better understand the behavior of the function. --- ### Step 1: Understand the function The function is: \[ f(x) = \frac{6}{x - 6} \] - As \(x \to 6^+\), \(x\) approaches 6 from **values greater** than 6. - The denominator \(x - 6\) approaches 0 from **positive** values. - The numerator is a constant 6. --- ### Step 2: Behavior near \(x = 6^+\) Let's examine the function's values for \(x\) slightly greater than 6: | \(x\) | \(x - 6\) | \(f(x) = \frac{6}{x - 6}\) | |-------------|-----------|----------------------------| | 6.1 | 0.1 | \(\frac{6}{0.1} = 60\) | | 6.01 | 0.01 | \(\frac{6}{0.01} = 600\) | | 6.001 | 0.001 | \(\frac{6}{0.001} = 6000\) | | 6.0001 | 0.0001 | \(\frac{6}{0.0001} = 60000\) | **Observation:** - As \(x\) approaches 6 from the right, \(f(x)\) becomes larger and larger, tending towards infinity. --- ### Step 3: Graphical interpretation Plotting the function \(f(x) = \frac{6}{x - 6}\): - For \(x > 6\), the graph starts just above the vertical asymptote at \(x=6\). - As \(x \to 6^+\), the graph rises sharply towards \(+\infty\).  *(Note: Visualize a hyperbolic curve approaching the vertical asymptote at \(x=6\) from the right)* --- ### Step 4: Conclusion based on tables and graph - The table shows that as \(x\) gets closer to 6 from the right, \(f(x)\) increases without bound. - The graph confirms that the function tends to \(+\infty\) as \(x \to 6^+\). --- ### **Final answer:** \[ \boxed{ \lim_{x \to 6^+} \frac{6}{x - 6} = +\infty } \] **Summary:** Using tables and graphs, we've shown that the function's value increases without limit as \(x\) approaches 6 from the right, leading to the conclusion that the limit is positive infinity.