Problem 8 - Calculus Single Variable

Question

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \begin{array}{l} \lim _{x \rightarrow-1} f(x) \text { for } \\ \qquad f(x)=\left\\{\begin{aligned} 6 & \text { if } x \leq-1 \\ 10 & \text { if } x>-1 \end{aligned}\right. \end{array} $$

Step-by-Step Solution

Answer

The limit does not exist because the left-hand limit and the right-hand limit are different.

Step 1: Analyze the limit from the left

To analyze the limit as \(x\) approaches \(-1\) from the left, we check the value of \(f(x)\) for \(x\leq -1\). Given, \(f(x) = 6\) for \(x \leq -1\), hence the left-hand limit \(\lim_{x \to -1^-} f(x) = 6\).

Step 2: Analyze the limit from the right

To find the right-hand limit, analyze \(f(x)\) as \(x\) approaches \(-1\) from the right, meaning \(x > -1\). Here, \(f(x) = 10\). Thus, the right-hand limit \(\lim_{x \to -1^+} f(x) = 10\).

Step 3: Compare the left and right limits

For the overall limit \(\lim_{x \to -1} f(x)\) to exist, the left-hand limit and the right-hand limit must be equal. Here, \(\lim_{x \to -1^-} f(x) = 6\) and \(\lim_{x \to -1^+} f(x) = 10\), which are not equal. Thus, the limit does not exist.

Key Concepts

Piecewise Functions

Piecewise functions are intriguing types of functions where different rules or formulas are applied to different intervals of the independent variable, usually denoted as \(x\). These functions are commonly used to represent situations where a variable behaves differently in separate ranges. A classic example includes tax brackets or shipping rates, where different policies apply for different income levels or weight ranges.

In our given problem, we encounter a piecewise function defined as follows:

• For \(x \leq -1\), the function takes the value \(6\).
• For \(x > -1\), the function takes the value \(10\).

This setup means that depending on whether \(x\) is less than or equal to, or greater than \(-1\), the function \(f(x)\) outputs a different value.

These kinds of functions are pivotal in calculus when we start discussing limits, as approaching a specific point from different directions could yield different results. A well-understood piecewise function allows us to dissect and analyze each interval independently, which is crucial for solving limits problems.

Left-hand Limit

The left-hand limit focuses on what happens to a function as \(x\) approaches a specific point from the left side. Imagine standing to the left of \(x = -1\) and moving towards it. This journey from the left helps us understand the behavior of the function specifically on that side.

In our piecewise function, when looking at \(x\) approaching \(-1\) from the left, we only need to consider the range where \(x \leq -1\). Therefore, we focus on the part of the piecewise function where \(f(x) = 6\). Hence, as we stride closer to \(-1\), the values of \(f(x)\) stabilize at \(6\). Formally, we write this as \(\lim_{x \to -1^-} f(x) = 6\).

Understanding the left-hand limit assists us in determining whether the function is consistent as we approach \(x = -1\) from this side, which is fundamental for identifying if an overall limit exists at this point.

Right-hand Limit

The right-hand limit is an examination into what happens when \(x\) draws nearer to a specific point but approaches it from the right side. It's like standing to the right of \(x = -1\) and stepping towards \(-1\). This side-specific look is just as crucial because piecewise functions can act differently on either side of a point.

For our function \(f\), when observing \(x\) as it approaches \(-1\) from the right, we look at where \(x > -1\). In this interval, \(f(x) = 10\). Thus, moving closer to \(-1\) from this direction aligns \(f(x)\) with the value \(10\). We denote this behavior formally as \(\lim_{x \to -1^+} f(x) = 10\).

Knowing the right-hand limit provides clarity about the output values of the function when coming towards \(x = -1\) from this side. By identifying this limit, we gain insight into how smooth or jumpy the function might be around the point of interest. Conclusively, if left and right limits differ, as they do here (since \(6eq10\)), the overall limit at that point does not exist.