91Ó°ÊÓ

The concept of a limit is fundamental in calculus and helps us understand the behavior of functions as they approach specific points. Specifically, when we say \( \lim_{x \to a} f(x) = L \), we mean that as \(x\) gets closer and closer to \(a\), the values of the function \(f(x)\) approach the number \(L\).

For example, in the exercise \(\lim _{x \rightarrow 0} f(x) \) where \(f(x) = |x - 2|\), we are interested in the values that \(f(x)\) approaches as \(x\) approaches 0, not the function's actual value when \(x\) is exactly at 0. The limit here is 2 because \(f(x)\) approaches the value of 2 as \(x\) gets nearer to 0. Understanding the definition of limits allows students to better grasp the concept of approaching values, rather than just evaluating a function at a point.

The absolute value of a real number is a measure of its distance from zero on the number line, regardless of direction. As such, absolute value is always non-negative. The formal definition of the absolute value of \(x\) is \(|x| = \{ x, \text{if } x \geq 0; -x, \text{if } x < 0 \}\).

In the context of the exercise, the function \(f(x) = |x - 2|\) gives us the distance of \(x\) from the number 2. As an example, regardless if \(x\) is slightly more or less than 2, after applying the absolute value, the result is a positive distance. Therefore, when finding limits of absolute value functions, it is crucial to consider both sides of the point you are approaching. The properties of absolute value must be remembered, in particular, that the output of an absolute value is always non-negative, which is why \(\lim_{x \rightarrow 0} f(x) = 2\) and not -2.

Continuity of a function at a point means that there are no breaks, jumps, or holes at that point in the graph of the function. A function is continuous at a point \(a\) if \(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)\).

This means that the left-hand limit (as you approach \(a\) from the left) and the right-hand limit (as you approach \(a\) from the right) are equal and they both equal the value of the function at \(a\). In our exercise, the function \(f(x) = |x - 2|\) is continuous at all points on the real line, including at \(x = 2\). Both sides approach the same value as \(x\) approaches 2, and the value of the function at \(x = 2\) is 0. Therefore, \(\lim_{x \rightarrow 2} f(x) = 0\), which is consistent with the continuous nature of an absolute value function.