91Ó°ÊÓ

When dealing with limits, one of the first steps is often to simplify the function. This can make evaluating the limit much easier and clearer. In our specific problem, the function is given as:

• \(f(x) = \frac{(x-1)^2}{(x-1)^2}\)

To simplify, notice that the numerator and the denominator are identical for any \(x\) except \(x = 1\). Therefore, we can cancel these terms out, provided we remember that this simplification does not hold true at the point where the division by the denominator equals zero. So, the simplified function becomes:

• \(f(x) = 1\) for all \(x eq 1\)

Remember, simplification is key in calculating limits efficiently. It's often the case that such simplifications can transform a complex expression into something that's much easier to work with, even if it’s not valid for the troublesome point.

The process of limit evaluation is concerned with finding the value that a function approaches as the input approaches a particular point. In our example, after simplification, the function is:\
\(f(x) = 1\) for all \(x eq 1\). This means that no matter how close \(x\) gets to 1, the function value remains 1. Thus, when we evaluate the limit:

• \(\lim_{x \rightarrow 1} f(x) = 1\)

This result tells us that even if \(x\) cannot be exactly 1 because of the original formulation causing potential division by zero, the behavior of \(f(x)\) as \(x\) gets closer to 1 still consistently stays at 1. Limit evaluation often requires us to use substitution, simplification, or L’Hôpital's Rule if the limits evaluate to indeterminate forms like \(\frac{0}{0}\). Here, simple cancellation solves the indeterminate form, allowing us to directly calculate the limit.

Division by zero occurs in situations where the denominator in a fraction becomes zero. It can lead to undefined behavior or what is called an indeterminate form. In our exercise, the function before simplification is:

• \(f(x) = \frac{(x-1)^2}{(x-1)^2}\)

At \(x = 1\), evaluating this directly would lead to \(\frac{0}{0}\), an indeterminate form. To handle this type of problem:

• Identify where the function may become undefined by setting the denominator equal to zero.
• Simplify the expression as much as possible to see if the problematic factor can be canceled.
• Reevaluate the limit after simplification, ensuring the terms that cause division by zero are appropriately managed.

Division by zero is something that calculators might not handle well, but in calculus, with careful simplification and understanding of limits, such problems can typically be resolved analytically.