Problem 61 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ \begin{array}{l} F(x)=\left\\{\begin{array}{ll} 2 x+1, & \text { for } x<1 \\ x, & \text { for } x \geq 1 \end{array}\right. \\ \text { Find } \lim _{x \rightarrow 1^{-}} F(x), \lim _{x \rightarrow 1^{+}} F(x), \text { and } \lim _{x \rightarrow 1} F(x) . \end{array} $$

Step-by-Step Solution

Answer

\( \lim_{x \to 1^{-}} F(x) = 3 \), \( \lim_{x \to 1^{+}} F(x) = 1 \), \( \lim_{x \to 1} F(x) \) does not exist.

Step 1: Analyze the piecewise function

The given function is piecewise defined: \( F(x) = 2x + 1 \) for \( x < 1 \) and \( F(x) = x \) for \( x \geq 1 \). This means that the function has two different expressions depending on the value of \( x \).

Step 2: Graph the function

Create a graph with two parts: one line for \( x < 1 \) (using \( F(x) = 2x + 1 \)) and another for \( x \geq 1 \) (using \( F(x) = x \)). The graph of \( F(x) = 2x + 1 \) is a line with slope 2, intersecting the y-axis at 1, which ends at \( x = 1 \) as an open circle. For \( x \geq 1 \), the graph of \( F(x) = x \) starts with a closed circle at \( x = 1 \) and follows a diagonal line.

Step 3: Determine \( \lim_{x \to 1^{-}} F(x) \)

For \( x \to 1^{-} \), the relevant part of the function is \( F(x) = 2x + 1 \). Compute the limit: \( \lim_{x \to 1^{-}} F(x) = 2(1) + 1 = 3 \).

Step 4: Determine \( \lim_{x \to 1^{+}} F(x) \)

For \( x \to 1^{+} \), the function is \( F(x) = x \). Compute the limit: \( \lim_{x \to 1^{+}} F(x) = 1 \).

Step 5: Determine \( \lim_{x \to 1} F(x) \)

Since \( \lim_{x \to 1^{-}} F(x) eq \lim_{x \to 1^{+}} F(x) \), the limit \( \lim_{x \to 1} F(x) \) does not exist.

Key Concepts

Piecewise Functions

Piecewise functions are special mathematical functions defined by multiple distinct expressions over separate intervals of a variable. These intervals are typically determined by the value of 'x', and each section of the piecewise function defines what the equation looks like in that specific part.

In the exercise given, the function \( F(x) \) consists of two separate expressions:

• For values of \( x < 1 \), \( F(x) = 2x + 1 \).
• For values of \( x \geq 1 \), \( F(x) = x \).

Such functions are commonly used in mathematics to model real-world situations where a process undergoes changes at distinct points.

When working with piecewise functions, it's pivotal to clearly understand how each part operates within its domain. This includes looking at endpoints where expressions change, like at \( x = 1 \) in this example. It’s also essential to understand the continuity of the function, which can be observed by examining limit behavior as 'x' approaches critical points from both sides.

Graphing Functions

Graphing functions is crucial for visualizing how a piecewise function behaves over different segments of its domain. To accurately graph a piecewise function, each piece should be drawn over its respective interval, ensuring to represent open or closed points appropriately at boundaries.

For the provided function \( F(x) \), the graph consists of:

• A line representing \( 2x + 1 \) for \( x < 1 \). This line has a slope of 2 and intersects the y-axis at 1. \( x = 1 \) is an open circle, indicating \( 2 imes 1 + 1 = 3 \) isn't included.
• The line \( x \) for \( x \geq 1 \) starts at \( x = 1 \) with a closed circle, continuing diagonally with a slope of 1.

Graphing helps in visually assessing where and how limits at specific points, such as at boundaries, should be evaluated. It also aids in identifying discontinuities or hops in the function's behavior.

One-Sided Limits

One-sided limits offer insight into the tendency or behavior of a function as 'x' approaches a particular point from one direction specifically. Calculating these is crucial when working with piecewise functions, especially at points where function expressions change.

In the exercise:

• We calculate \( \lim_{x \to 1^{-}} F(x) \) to find the function's behavior as it reaches '1' from the left. Given \( F(x) = 2x + 1 \) for \( x < 1 \), this limit equals 3.
• For \( x \to 1^{+} \), \( F(x) = x \) is used. Thus, \( \lim_{x \to 1^{+}} F(x) = 1 \).

It's crucial to note that when these one-sided limits don't match, as in this case, the overall limit \( \lim_{x \to 1} F(x) \) does not exist, signifying a discontinuity at that point. Understanding one-sided limits helps predict whether there is a jump or a sharp corner in a function's graph.