91Ó°ÊÓ

Continuity is a fundamental concept in calculus and analysis, which describes a condition where small changes in the input of a function lead to small changes in the output. For a function to be continuous at a point, specifically at \(x = c\), three conditions must be satisfied:

• The function \(f(x)\) must be defined at \(x = c\).
• The limit \(\lim_{x \to c} f(x)\) must exist.
• The limit must be equal to the function's value: \(f(c) = \lim_{x \to c} f(x)\).

When these conditions are met, the function can be graphed without any breaks or gaps at that point. Continuous functions exhibit predictable behaviors which simplifies the process of analyzing them mathematically. In this exercise, the expression \((x^2 - 1)(2 - \cos x)\) was continuous at \(x = 0\), allowing us to evaluate the limit via direct substitution without complexity. This is because both components, \(x^2 - 1\) and \(2 - \cos x\), are continuous functions and retain their continuity when multiplied together.

The substitution method allows us to find limits by directly replacing the variable with its limiting value. This method can be used when a function is continuous at the point we're evaluating. In our exercise, to find \( \lim_{x \to 0} (x^2 - 1)(2 - \cos x)\), we substituted \(x = 0\) into the expression:

• The term \(x^2 - 1\) becomes \(0^2 - 1 = -1\).
• The term \(2 - \cos x\) becomes \(2 - \cos(0) = 2 - 1 = 1\).

Combining these results gives \((-1)(1) = -1\). The substitution method works seamlessly when the conditions of continuity are satisfied, making it a straightforward approach to solve problems involving limits.

Evaluating limits often involves various techniques that ensure we accurately measure a function's behavior as it approaches a specific point. Some key techniques include:

• **Direct Substitution:** Used primarily for continuous functions, where the variable is directly replaced with the limiting value, like in our example with \(x = 0\).
• **Factoring and Simplification:** These techniques handle more complex expressions by reducing them to simpler forms before evaluating.
• **Rationalization:** Commonly used for expressions involving roots, this technique involves multiplying by the conjugate to simplify.
• **L'Hôpital's Rule:** This powerful tool applies to indeterminate forms like \(0/0\) or \(\infty/\infty\), using derivatives to find the limit.

Our specific exercise did not require advanced techniques beyond direct substitution due to the continuity of the function. However, understanding these methods is fundamental for tackling a wide range of limit problems in calculus, providing students with a comprehensive toolbox for problem-solving.