Problem 51 - College Algebra and Calculus: An Applied Approach

Question

$$ \lim _{x \rightarrow 2} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} 4-x, & x \neq 2 \\ 0 & x=2 \end{array}\right. $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches 2 of the function \(f(x)\) is 2.

Step 1: Understand the function

First, you must understand that the given function f(x) is a piecewise function that is defined differently for x ≠ 2 and x = 2. However, the limit as x approaches 2 depends on the values of the function as x gets closer to 2 from both sides (left and right), not on the value of the function exactly at x = 2.

Step 2: Apply the limit to the function

The limit as x approaches 2 of f(x) is evaluated by substituting x = 2 in the function defined for x ≠ 2 since the function's value at x = 2 does not matter. Here, f(x) is defined as 4 - x for x ≠ 2.

Step 3: Compute the limit

By substituting \(x = 2\) in \(4 - x\), the result is \(4 - 2 = 2\). Therefore, the limit of the function as \(x\) approaches 2 is 2.

Key Concepts

Limits of Functions

In calculus, a limit is a fundamental concept that describes the value a function approaches as the input nears a certain point. Understanding limits is essential as it lays the groundwork for derivatives and integrals. When we're asked to find the limit of a function, we're trying to understand how the function behaves as the input gets very close to a particular number.

When evaluating limits, we often use the notation like \( \lim_{x \to a} f(x) \), which means we're finding the limit of \( f(x) \) as \( x \) approaches \( a \). It is crucial to note:

• Limits assess the behavior of a function as it nears a specific point but do not necessarily involve evaluating the function at that point.
• The actual value at the point may differ or might even be undefined; what matters is the trend or the pattern of the outputs as the inputs get very close to the specified number.

This concept is particularly important when dealing with continuous and discontinuous functions.

Piecewise Functions

Piecewise functions are fascinating because they are defined by different expressions based on the input's value. They can behave differently in various segments of their domain, making them versatile for modeling real-world situations. The function provided in the original exercise, for instance, is a piecewise function.

A piecewise function can be represented as follows:

• For certain values of \( x \), one part of the function is applied (e.g., \( 4 - x \) when \( x eq 2 \)).
• For other values, another expression or a single value is used (e.g., output is \( 0 \) when \( x = 2 \)).

These functions are useful for solving limits because they often show us how the function behaves on either side of the point of interest. For the function in the exercise, the limit depends on \( x \) approaching 2, observing the behavior of the function defined by \( 4 - x \). This demonstrates that even if the function value at the exact point is different, it does not affect the limit.

Evaluating Limits

Evaluating limits of piecewise functions involves careful analysis. We need to look at the section of the function that affects the limit. In the given exercise, the limit as \( x \rightarrow 2 \) seeks to find the value \( f(x) \) approaches as \( x \) gets very close to 2.

Here's a simple process on how to approach evaluating limits:

• Identify the relevant part of the piecewise function. For \( x \to a \), use the function part where \( x eq a \).
• Substitute or simplify the expression to find what happens as \( x \) gets closer to the point. Ignore the value of the function at the point since it's not influencing the limit.

In the exercise, the relevant expression is \( 4-x \). By substituting \( x = 2 \) in this equation, we find that the limit approaches 2, demonstrating that the function value \( f(2) \) being 0 does not affect the limit calculation. Evaluating limits becomes intuitive with practice and careful observation of each piecewise segment's behavior in relation to the approach point.