Problem 59 - Precalculus with Limits

Question

In Exercises 55-62, graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. \\[ \lim_{x \to 2} f(x)\ \textrm{where}\ f(x) = \begin{cases} x-1, & \quad x\leq2\\\ 2x-3, & \quad x>2 \end{cases} \\]

Step-by-Step Solution

Answer

\(\lim_{x \to 2} f(x) = 1\)

Step 1: Graph the Piecewise Function

The function given is a piecewise function. That means it has different formulas for different intervals of x. For \(x \leq 2\), the function is \(f(x) = x - 1\). For \(x > 2\), the function is \(f(x) = 2x - 3\). Plot these two functions on the same graph, with a vertical dividing line at \(x = 2\).

Step 2: Evaluate the Left-hand Limit

For \(x \leq 2\), use the formula \(f(x) = x - 1\). Evaluate the function at \(x = 2\) to get \( \lim_{x \to 2^-} f(x) = 2 - 1 = 1\). This is the left-hand limit.

Step 3: Evaluate the Right-hand Limit

For \(x > 2\), use the formula \(f(x) = 2x - 3\). Evaluate the function at \(x = 2\) to get \( \lim_{x \to 2^+} f(x) = 2*2 - 3 = 1\). This is the right-hand limit.

Step 4: Determine the Overall Limit

Compare the left-hand and right-hand limits. If they are equal, that value is the overall limit. In this case, both are equal to 1, therefore, \( \lim_{x \to 2} f(x) = 1\)

Key Concepts

One-Sided Limits

When analyzing piecewise functions, it's crucial to evaluate limits from both directions - left and right. These are known as one-sided limits. By finding these limits, you can determine the behavior of a function at a specific point.

• For the left-hand limit, which is denoted as \(\lim_{x \to a^-} f(x)\), we approach the point \(a\) from values less than \(a\).
• For the right-hand limit, or \(\lim_{x \to a^+} f(x)\), we approach \(a\) from values greater than \(a\).

Let's consider an example. For the piecewise function \(f(x)\), determine \(\lim_{x \to 2} f(x)\). When \(x\) is approaching 2 from the left, use the formula \(f(x) = x - 1\). Here, \(\lim_{x \to 2^-} f(x) = 1\). From the right, use \(f(x) = 2x - 3\), giving \(\lim_{x \to 2^+} f(x) = 1\). Both sides coming to the same value tells us about the continuity or breakpoint at that specific \(x\).

Graphing Piecewise Functions

Graphing piecewise functions can seem tricky, but breaking it into steps makes it manageable. Here's how you can effectively graph them:

• Identify each segment of the piecewise function by its specific interval and formula.
• Plot each part on the graph, ensuring you follow the interval guidance.
• Mark divisions on the x-axis where the function formula changes.
• Pay attention to open and closed dots on graphs to signify where a point is included or excluded.

Consider the given example: - For \(x \leq 2\), graph \(f(x) = x - 1\). Your graph will be a straight line starting from a y-value of -1 with a slope of 1.
- For \(x > 2\), graph \(f(x) = 2x - 3\) which starts at the point where \(x\) is slightly more than 2.
The point \(x = 2\) becomes a vertical dividing line. Graphed properly, piecewise functions present a comprehensive view of how functions behave in their respective domains.

Calculus Limits

Calculus limits provide a fundamental idea of what a function's behavior is as it approaches a particular point. The general notation for a limit as \(x\) approaches a value \(a\) is \(\lim_{x \to a} f(x)\).

• Limits are crucial for defining derivatives and integrals, leading to a deeper understanding of calculus concepts.
• Understanding limits helps in determining function continuity, smoothness of curves, and discovering whether they connect or not.

In our exercise, when both the left-hand and right-hand limits are calculated as 1, the overall limit \(\lim_{x \to 2} f(x) = 1\). This means that as \(x\) nears 2 from either side, \(f(x)\) is tending toward the value of 1. Calculating these matching limits ensures the function doesn't have any sudden jumps at \(x = 2\), informing us that \(f(x)\) behaves consistently around this point.