Question

Approximate the given limits both numerically and graphically. $$ \begin{array}{l} \lim _{x \rightarrow 2} f(x), \text { where } \\ f(x)=\left\\{\begin{array}{cl} x+2 & x \leq 2 \\ 3 x-5 & x>2 \end{array}\right. \end{array} $$

Short answer

The limit does not exist because the left and right limits differ.

Step-by-step solution

Understand the function

The given function is a piecewise function with two parts. For values of \( x \) less than or equal to 2, the function is \( f(x) = x + 2 \). For values of \( x \) greater than 2, the function is \( f(x) = 3x - 5 \). Our task is to find the limit of \( f(x) \) as \( x \) approaches 2.

Evaluate the limit from the left (\( x \to 2^- \))

For \( x \leq 2 \), use \( f(x) = x + 2 \). As \( x \) approaches 2 from the left, substitute \( x = 2 \) in the equation: \( f(x) = 2 + 2 = 4 \). Thus, the left-hand limit is 4.

Evaluate the limit from the right (\( x \to 2^+ \))

For \( x > 2 \), use \( f(x) = 3x - 5 \). As \( x \) approaches 2 from the right, substitute \( x = 2 \) in the equation: \( f(x) = 3(2) - 5 = 1 \). Thus, the right-hand limit is 1.

Compare the limits

We found the left-hand limit to be 4 and the right-hand limit to be 1. Since these two values are different, the limit \( \lim_{x \to 2} f(x) \) does not exist.

Graphical Approximation

Graph the function \( f(x) \) using a graphing tool or by hand. You will plot the line \( y = x + 2 \) for \( x \leq 2 \) and the line \( y = 3x - 5 \) for \( x > 2 \). The graph will show a jump at \( x = 2 \), confirming that the left and right limits are different, hence the limit does not exist.

Key concepts

Piecewise Functions

A piecewise function is a function composed of multiple sub-functions, each defined on a specific interval within the domain. Such functions are often described with different rules depending on the value of the input. For instance, in our problem, the function \( f(x) \) is defined as:

• \( f(x) = x + 2 \) when \( x \leq 2 \)
• \( f(x) = 3x - 5 \) when \( x > 2 \)

This structure allows the function to have different behaviors depending on the interval of the input. Understanding how to work with piecewise functions is crucial, as they can be commonly found in real-world applications where conditions change across different scenarios. By analyzing each piece separately, we can determine the overall behavior of the function.

Left-Hand Limit

The left-hand limit of a function as \( x \) approaches a specific value is what the function's values approach from the left side of that value. For our piecewise function, we are interested in \( x \to 2^- \), which means looking at values just less than 2.
For \( x \leq 2 \), we use the function \( f(x) = x + 2 \). As \( x \) gets closer to 2 from the left, we substitute \( x = 2 \) into the equation, getting \( f(2) = 2 + 2 = 4 \).
Therefore, the left-hand limit of \( f(x) \) as \( x \to 2 \) is \( 4 \). Evaluating left-hand limits helps us understand how the function behaves near the boundary of its intervals.

Right-Hand Limit

The right-hand limit investigates what the function's values approach from the right side of a given point. In this exercise, our focus is on \( x \to 2^+ \), considering values slightly more than 2.
For \( x > 2 \), our function is \( f(x) = 3x - 5 \). As \( x \) approaches 2 from the right, we calculate \( f(2) = 3 \times 2 - 5 = 1 \).
This means the right-hand limit of \( f(x) \) as \( x \to 2 \) is \( 1 \). Finding right-hand limits gives insight into the function's behavior and helps identify any discontinuities or jumps at particular points.

Graphing Functions

Graphing functions provides valuable insight into their behavior across different intervals. For our piecewise function, we have two distinct parts that need to be graphed separately:

• Plot \( y = x + 2 \) for \( x \leq 2 \)
• Plot \( y = 3x - 5 \) for \( x > 2 \)

When graphed, these two parts show a clear difference at \( x = 2 \). The line \( y = x + 2 \) stops at \( (2,4) \), while the line \( y = 3x - 5 \) begins at \( (2,1) \), creating a visible gap or jump.
This visual discrepancy confirms the non-existence of a unified limit at \( x = 2 \). Graphing functions, especially piecewise ones, aids in visualizing dynamics and pinpointing abrupt changes in behavior, integral to understanding limit concepts.