91Ó°ÊÓ

Graphical analysis is a powerful tool in calculus, especially for understanding limits. By examining the function visually, we can gain insights into how the function behaves as the variable approaches a particular value. In this exercise, you are tasked with graphing the function \(\frac{\frac{1}{x} - 1}{1-x}\) which simplifies to \(\frac{1}{x}\). By plotting \(\frac{1}{x}\), we can observe its behavior as it nears \(x = 1\).
The TRACE function on a graphing calculator is particularly helpful. It allows you to move along the curve and see what happens at specific points, including those very close to \(x = 1\). A smooth curve that appears to approach a specific value signals that the limit exists at that point.
Alternatively, you can use the TABLE feature, which displays function values at predefined intervals around \(x = 1\). This shows how \(y\), the output of the function, changes in response to changes in \(x\). Both these methods offer a clear picture of the limit and provide a visual confirmation of mathematical calculations.

Evaluation of a limit is a fundamental step in calculus. It involves finding what value a function approaches as the input approaches a certain number. In our original problem, we need to evaluate \(\lim _{x \rightarrow 1} \frac{\frac{1}{x} - 1}{1-x}\). By simplifying this expression, we end up computing \(\lim _{x \rightarrow 1} \frac{1}{x}\).
As you approach \(x = 1\), you can try values closer and closer to \(x = 1\), such as \(0.9, 0.99,\) and \(1.01\). These numbers provide approximations that, when plugged into \(\frac{1}{x}\), result in values that get very near to \(1\).
The goal is to see that as \(x\) gets increasingly close to \(1\), the output of the function approaches a specific number, which in this case is \(1\). This method of substituting values and observing outputs helps us confirm that the limit of \(\frac{1}{x}\) as \(x\) approaches \(1\) is indeed \(1\).