91Ó°ÊÓ

Direct substitution is often the first step in limit evaluation. It involves replacing the variable in the function with the value it is approaching. This method is straightforward and is used when the function is continuous at the point in question. For the given exercise, we directly substituted the value 2 into the expression:\[ \frac{x^3-6x-2}{x^3+4} \]Calculating directly, we replaced every occurrence of \(x\) with 2:\[ \frac{2^3 - 6(2) - 2}{2^3 + 4} = \frac{8 - 12 - 2}{8 + 4} = \frac{-6}{12} \]Direct substitution worked because there was no division by zero or indeterminate form. If the substitution results in a real number, the limit is simply that number.

In mathematics, simplifying fractions is a fundamental skill that helps in understanding and calculating with fractions more easily. After direct substitution in the exercise, we obtained the fraction:\[ \frac{-6}{12} \]This fraction can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 6. Simplification steps are as follows:- Divide \(-6\) by 6 to get \(-1\).- Divide 12 by 6 to get 2.Thus, the simplified form of the fraction is:\[ \frac{-1}{2} \]Simplifying fractions not only provides a cleaner solution but also allows for better interpretation of results. If a fraction can be reduced, always simplify it to its lowest terms.

Continuity in calculus refers to a function being seamless around a certain point, meaning that it has no gaps, jumps, or holes in its graph at that point. For functions where the limit can be evaluated through direct substitution, as in the exercise:\[ \lim_{x \to 2} \frac{x^3-6x-2}{x^3+4} \]If the limit exists and equals the value of the function at that point, then the function is continuous at that point. Here, substituting \(x = 2\) directly gives a real number, indicating the function is defined at \(x = 2\) and is continuous around this value.More formally, a function \(f(x)\) is continuous at a point \(c\) if:- The function is defined at \(c\).- The limit exists as \(x\) approaches \(c\).- The limit equals \(f(c)\).Understanding continuity helps in knowing when direct substitution methods can be applied effectively, ensuring accurate limit calculations.