Problem 28 - Precalculus with Limits

Question

In Exercises 27 and 28, graph the function and find the limit(if it exists) as \(x\) approaches 2. \\[ f(x)= \left\\{ \begin{array}{rr} 2x,& \mbox{if \(x \leq 2\)};\\\ x^2 -4x+1,& \mbox{if \(x > 2\)}.\end{array} \right. \\]

Step-by-Step Solution

Answer

The graph of the function consists of a line segment and a portion of a parabola. The limit as \(x\) approaches 2 does not exist because the limits from the left and right are different (4 and -3, respectively).

Step 1: Graph the function

The given function is a piecewise function with different definitions for \(x \leq 2\) and \(x > 2\). On the left part of the graph, a straight line defined by \(2x\) is drawn for \(x \leq 2\). On the right side, a quadratic function defined by \(x^2 -4x+1\) is drawn for \(x > 2\).

Step 2: Find the limit from the left side

The limit from the left, denoted \(\lim_{x \to 2^-} f(x)\), is computed using the piece of the function defined for \(x \leq 2\), which is simply \(2x\). Substituting 2 for \(x\) gives \(2(2)\), which equals 4.

Step 3: Find the limit from the right side

The limit from the right, denoted \(\lim_{x \to 2^+} f(x)\), is computed using the piece of the function defined for \(x > 2\), which is \(x^2 -4x+1\). Substituting 2 for \(x\) gives \(2^2 - 4(2) + 1\), which equals -3.

Step 4: Determine if the limit exists

Since the limits from the left and right are different (\(4 \neq -3\)), the overall limit does not exist. If the two limits were the same, then the overall limit would exist and be equal to that common value.

Key Concepts

Piecewise Functions

Piecewise functions are mathematical expressions that have different definitions or expressions for different intervals of the input variable. Here, we deal with a piecewise function defined as:

• For values of \(x\) less than or equal to 2, the function is defined by \(f(x) = 2x\).
• For values of \(x\) greater than 2, the function changes and is defined by \(f(x) = x^2 - 4x + 1\).

This means depending on the value of \(x\), the function will take on a different form, either linear or quadratic. Note that the transition point or 'breaking point' at \(x = 2\) is crucial in these functions as it marks where the definition changes. Piecewise functions are commonly used to model situations where different conditions apply in different scenarios.

Graphing

Graphing a piecewise function involves plotting each segment over its specified range.

• For \(x \leq 2\), plot the linear equation \(y = 2x\), which is a straight line with a slope of 2.
• For \(x > 2\), plot the quadratic equation \(y = x^2 - 4x + 1\), which forms a parabola opening upwards.

The key to graphing such functions effectively is to clearly mark the changeover point, which is \(x = 2\) in this case. Here, the line and the parabola meet but are not connected as the function has different values from the left and right. When graphing, use solid or open dots to indicate whether the function includes the endpoint or not, providing a clear visual distinction.

Right-hand Limit

The right-hand limit refers to the value that a function approaches as the input approaches a certain point, from the right side. In our example, the right-hand limit at \(x = 2\) is found by evaluating the function using the expression defined for \(x > 2\), which is \(x^2 - 4x + 1\). Calculation involves substituting \(x = 2\) and results in:\[2^2 - 4(2) + 1 = -3\]Therefore, the right-hand limit of the function as \(x\) approaches 2 is \(-3\). It’s important to understand this concept because it shows how the function behaves from a specific direction, which can differ from its behavior from the other side.

Left-hand Limit

The left-hand limit is the value a function approaches as the input approaches a certain point from the left-hand side. This involves using the expression defined for \(x \leq 2\), which in our scenario is \(f(x) = 2x\).To find the left-hand limit, we substitute \(x = 2\) into the equation, giving\(2(2) = 4\).Thus, the left-hand limit of the function at 2 is \(4\). By calculating left and right-hand limits, we determine whether a single, overall limit exists at a point. Here, since the left-hand and right-hand limits are not equal (4 vs -3), there’s no singular limit at \(x = 2\). This analysis provides insight into the continuity and behavior of functions at specified points.