Question

Use a calculator to graph the function and estimate the value of the limit, then use L'Hôpital's rule to find the limit directly. $$ \lim _{x \rightarrow 1} \frac{(x-1)^{2}}{\ln x} $$

Short answer

The limit is 0.

Step-by-step solution

Graphing the Function

First, input the function \(f(x) = \frac{(x-1)^2}{\ln x}\) into a graphing calculator. Use the calculator to approximate the value of the limit as \(x\) approaches 1. You should notice that near \(x=1\), the function seems to approach a particular value.

Estimating the Limit Using the Graph

Based on the graph, observe the behavior of the function \(f(x)\) as \(x\) gets very close to 1. The graph should suggest that the limit is a small value close to 0, indicating that the limit is likely 0.

Confirming Indeterminate Form

Before applying L'Hôpital's Rule, confirm that \(\lim _{x \rightarrow 1} \frac{(x-1)^2}{\ln x}\) results in an indeterminate form \(\frac{0}{0}\). As \(x\) approaches 1, \((x-1)^2\) approaches 0 and \(\ln x\) approaches \(\ln 1 = 0\). Thus, the limit is an indeterminate form suitable for L'Hôpital's Rule.

Applying L'Hôpital's Rule

Apply L'Hôpital's Rule which states that for limits of indeterminate form \(\frac{0}{0}\), \(\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}\), if this limit exists. Calculate the derivatives: 1. The derivative of \((x-1)^2\) is \(2(x-1)\).2. The derivative of \(\ln x\) is \(\frac{1}{x}\).So, the limit becomes:\[\lim_{x\to 1}\frac{2(x-1)}{\frac{1}{x}} = \lim_{x\to 1} 2x(x-1)\].

Calculating the Limit

Find the limit \(\lim_{x\to 1} 2x(x-1)\). Substitute \(x = 1\), getting:\[2(1)(1-1) = 0\]. Thus, the limit evaluates to 0.

Key concepts

Limits

Limits are a fundamental concept in calculus, helping us understand how functions behave as their input values approach a certain point. Often, we want to find the limit of a function as it nears a specific point without necessarily reaching or being defined at that point. For example, in the exercise, the limit of the function \( \frac{(x-1)^2}{\ln x} \) as \( x \) approaches 1 is of interest.

Limits are essential because they form the foundational basis for other calculus concepts such as derivatives and integrals. When evaluating limits, especially those that do not result in straightforward values, it's helpful to learn different techniques and tools, such as graphing the function, to get an initial approximation before applying more precise methods. In our case, using a calculator to graph the function offers a visual understanding and estimation of the limit around \( x = 1 \). By observing the behavior of the graph near this point, we develop an intuition about the function's behavior and, possibly, its limit value.

Indeterminate Forms

Indeterminate forms arise when we attempt to evaluate a limit and end up with an expression like \(\frac{0}{0}\), \(\infty - \infty\), or \(\frac{\infty}{\infty}\), among others. These forms are 'indeterminate' because they do not immediately suggest a definite value or behavior. In the exercise, as \( x \) approaches 1, both the numerator \((x-1)^2 \) and the denominator \( \ln x \) approach 0, leading to the form \( \frac{0}{0} \).

To successfully evaluate limits involving indeterminate forms, we employ techniques such as L'Hôpital's Rule, which can simplify these forms into more solvable expressions. L'Hôpital's Rule specifically allows us to differentiate the numerator and denominator separately and then take the limit of the resulting fraction. This process helps bypass the indeterminate nature by analyzing the function's rates of change rather than its absolute values.

Graphing Functions

Graphing functions provides a visual way to understand complex mathematical behavior and is particularly helpful in estimating limits. By plotting the function \( f(x) = \frac{(x-1)^2}{\ln x} \) on a graphing calculator, we can see how the function behaves as \( x \) approaches 1.

With graphing:

• You can spot trends, such as if the function is approaching a particular value (i.e., the limit).
• It helps identify if and where functions have asymptotes, discontinuities, or specific behaviors.
• You get insight into how changes in \( x \) affect the function, providing a more profound understanding of the function's characteristics.

By visual inspection of the graph, we noticed that as \( x \) nears 1, the function appears to trend towards the limit value of 0. Graphing complements analytical methods like L'Hôpital's Rule by giving a clearer picture of what happens as a function approaches its limit point.