Problem 27 - Calculus: Early Transcendentals

Question

Estimating limits graphically and numerically Use a graph f \(f\) to estimate \(\lim f(x)\) or to show that the limit does not exist. Evaluate \(f(x)\) near \(x=a\) to support your conjecture. $$f(x)=\frac{x-2}{\ln |x-2|} ; a=2$$

Step-by-Step Solution

Answer

Answer: The limit of the function as \(x\) approaches 2 is 0.

Step 1: Determine the left-hand limit by first finding the limit of the function as x approaches 2 from the left.

$$\lim_{x \to 2^-} \frac{x-2}{\ln|x-2|} = \lim_{x \to 2^-} \frac{x - 2}{\ln(x - 2)}$$ As \(x\) approaches 2 from the left, then \(x-2\) approaches 0 from the left, which means the logarithm approaches negative infinity, since \(\ln(x)\) approaches \(-\infty\) as \(x\) approaches 0 from the left. Hence, the left-hand limit of the function is: $$\lim_{x \to 2^-} \frac{x - 2}{\ln(x - 2)} = \frac{0}{-\infty} = 0$$

Step 2: Determine the right-hand limit by first finding the limit of the function as x approaches 2 from the right.

$$\lim_{x \to 2^+} \frac{x-2}{\ln|x-2|} = \lim_{x \to 2^+} \frac{x - 2}{\ln(x - 2)}$$ As \(x\) approaches 2 from the right (i.e., numbers greater than 2), \(x-2\) approaches 0 from the right. The logarithm therefore approaches \(+\infty\), since \(\ln(x)\) approaches \(\infty\) as \(x\) approaches 0 from the right. The right-hand limit becomes: $$\lim_{x \to 2^+} \frac{x - 2}{\ln(x - 2)} = \frac{0}{+\infty} = 0$$

Step 3: Compare the left-hand and right-hand limits.

Both the left-hand and right-hand limits are equal (0), which means that the limit of the function exists as \(x\) approaches 2. $$\lim_{x \to 2} f(x) = 0$$

Step 4: Estimate the limit graphically.

Plot the function \(f(x) = \frac{x - 2}{\ln |x - 2|}\) and observe its behavior as \(x\) approaches 2. The graph will show that, as x approaches 2, the value of \(f(x)\) approaches 0, confirming the limit we calculated.

Step 5: Estimate the limit numerically.

To further support the conjecture, evaluate the function near \(x = 2\). Choose points on both sides of \(a\) and calculate their corresponding function values. For example, select \(x = 1.99\) and \(x = 2.01\): $$f(1.99) = \frac{1.99-2}{\ln|1.99-2|} \approx -0.0183$$ $$f(2.01) = \frac{2.01-2}{\ln|2.01-2|} \approx 0.0187$$ As the values of \(x\) get closer to 2, the value of \(f(x)\) gets closer to 0. This confirms that the limit of the function as \(x\) approaches 2 is 0.

Key Concepts

Graphical Analysis

Graphical analysis is a valuable technique for estimating limits. By analyzing a graph of a function, we can visually inspect how a function behaves as it approaches a certain value. For the function \[f(x) = \frac{x-2}{\ln |x-2|}\] we want to estimate \[\lim_{x \to 2} f(x).\] To do this, we can plot the function on a graph and watch the path the function takes as the value of \(x\) gets closer to \(2\). - As \(x\) approaches \(2\), observe where the function values are heading. - Visual verification shows if the function stabilizes or converges to a certain value, in this case, zero. When you view the graph, you will notice that the plotted function approaches the value zero as \(x\) nears 2 from both the left and the right. This graphical check can reassure us about the limits calculated algebraically.

Numerical Analysis

In numerical analysis, we approximate limits by calculating values of a function very close to the point of interest. Let's take the function \(f(x) = \frac{x-2}{\ln |x-2|}\) near \(x = 2\). We can choose values a hairline away from 2, on both sides, to see where \(f(x)\) is heading. - Try values like \(x = 1.99\) and \(x = 2.01\). - Compute \(f(1.99) = \frac{1.99-2}{\ln|1.99-2|} \approx -0.0183\). - Compute \(f(2.01) = \frac{2.01-2}{\ln|2.01-2|} \approx 0.0187\). These results illustrate that as \(x\) gets closer to 2, \(f(x)\) converges towards 0. By listing these results in this way, we make a strong case for the existence and value of the limit.

One-sided Limits

One-sided limits help us understand how a function behaves as it gets close to a certain point from one direction. By investigating one-sided limits, we consider: - The left-hand limit (as \(x\) approaches a value from the left). - The right-hand limit (as \(x\) approaches the same value from the right). For \(f(x) = \frac{x - 2}{\ln|x - 2|}\), we calculate one-sided limits as \(x\) approaches 2: - **Left-hand Limit:** \[\lim_{x \to 2^-} \frac{x - 2}{\ln(x - 2)} = \frac{0}{-\infty} = 0.\] Here, both numerator and denominator approach zero and negative infinity, respectively, leading to zero. - **Right-hand Limit:**\[\lim_{x \to 2^+} \frac{x - 2}{\ln(x - 2)} = \frac{0}{+\infty} = 0.\] The numerator again approaches zero, and the denominator heads towards positive infinity, also resulting in zero. Since the left-hand and right-hand limits are equal, the two sides confirm that \(\lim_{x \to 2} f(x) = 0\) exists. This careful approach demonstrates that a function's behavior from different sides harmonizes to produce a single limit value.