91Ó°ÊÓ

Before calculating limits in calculus, expressions often need to be simplified. Factoring is a key step in this process, especially when dealing with polynomials. In the context of limits, factoring helps us handle expressions that might initially lead to undefined forms like division by zero.
To factor the expression in the denominator of our exercise, we start by recognizing that it’s a difference of squares. The original denominator is \( x^2 - 16 \), which can be rewritten as \((x-4)(x+4)\). This technique of expressing a difference of squares as a product of binomials is crucial because it may reveal common terms that can be canceled out, simplifying complex fractions and making the limit easier to calculate.

Finding limits is central to calculus, allowing us to determine the behavior of functions at specific points or as they approach infinity. In our exercise, we need to evaluate the limit \( \lim _{x \rightarrow 4^{-}} \frac{x^{2}}{x^{2} - 16} \).
After factoring, the expression becomes \( \frac{x^2}{(x-4)(x+4)} \). By simplifying the expression, we avoid the problem of dividing by zero. Once simplified, you substitute the value the variable is approaching. This can usually be done directly after simplification if there are no discontinuities or undefined points left.
The limit exists because after simplification, substituting \( x = 4 \) results in a defined value, bringing us closer to understanding the function’s behavior around the point.

Simplification of functions, especially when dealing with limits, involves reducing complex expressions into simpler forms. The goal is to make functions more workable, particularly when direct substitution leads to indeterminate forms like \( \frac{0}{0} \).
In our problem, by canceling common factors in the numerator and denominator, namely \( x \), the function \( \frac{x^2}{(x-4)(x+4)} \) is simplified to \( \frac{x}{x+4} \). This new expression is much easier to evaluate, especially when considering limits.
Through simplification, we ensure that there are no discontinuities caused by removable singularities, allowing us to substitute values directly and find the limit. This process not only helps in finding the limit but also provides insights into the continuity and differentiability of functions at given points.