91Ó°ÊÓ

When examining limits in calculus, a left-hand limit (also known as a "limit from the left") focuses on how a function behaves as the variable approaches a specific value from the left side. In more technical terms, for a function \(f(x)\) and a point \(a\), we denote the left-hand limit as \(\lim_{x \to a^-} f(x)\). Here, the minus sign indicates we are considering values of \(x\) slightly less than \(a\).

In the given exercise, we look at \(\lim_{x \to -8^-} f(x)\). These are values of \(x\) approaching a negative 8 from the left. Practically, it means we're considering numbers like -8.1, -8.01, -8.001, and so on. With the function \(f(x) = x^{2/3}\), even though \(x\) is negative, eventually, we take the cube root (which outputs a negative number) and then square it, resulting in a positive value.

Thus, for this function, as \(x\) values get closer to -8 from the left, \(f(x)\) consistently approaches 4. The left-hand limit, therefore, is 4.

Understanding right-hand limits, or "limits from the right," is similar to left-hand limits, but we look at the behavior of a function as it approaches from greater values. To express this, we use \(\lim_{x \to a^+} f(x)\), where the plus sign signifies approaching \(a\) from the right.

In our practice problem, analyze \(\lim_{x \to -8^+} f(x)\). Here, we examine \(x\) as it approaches negative 8 from slightly larger values. Think of examples like -7.9, -7.99, -7.999, closing in on -8. With \(f(x) = x^{2/3}\), when \(x\) is barely larger than -8, the cube root produces a negative result, but the squaring process turns it back positive.

This behavior ensures that just like the left-hand limit, the right-hand result approaches 4. Hence, as \(x\) approaches -8 from the right, the right-hand limit is found to also be 4.

The concept of a two-sided limit revolves around understanding what value a function approaches as the input comes from both directions—both from the left and the right. This is expressed as \(\lim_{x \to a} f(x)\). For this type of limit to exist, the left-hand limit must equal the right-hand limit at the point \(a\).

In this problem, after computing both left and right-hand limits as \(x\) approaches -8, we determined both were equal to 4. This equality confirms the existence of a two-sided limit, allowing us to conclusively state \(\lim_{x \to -8} f(x) = 4\).

In this scenario, because both one-sided limits matched, it showed the function's behavior was consistent as \(x\) neared -8 from any direction. We can then confidently say the two-sided limit confirms the function value the input approaches is truly 4.