Problem 17 - Calculus of a Single Variable

Question

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \begin{array}{ll}{\lim _{x \rightarrow 2} f(x)} & {} \\\ {f(x)=\left\\{\begin{array}{ll}{4-x,} & {x \neq 2} \\ {0,} & {x=2}\end{array}\right.}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the function

The given function is \(f(x) = 4-x\) for \(x \neq 2\) and \(0\) for \(x = 2\). This means that everywhere apart from \(x = 2\), the function simply behaves as \(y = 4 -x\).

Step 2: Substitute the limit

The limit as \(x\) approaches \(2\) of \(f(x)\) is then the same as the point on \(4 - x\) for \(x = 2\). Therefore, substituting, we get \(4-2 = 2\).

Step 3: Answer point

The limit of \(f(x)\) as \(x\) approaches \(2\) is \(2\). This indicates that as \(x\) gets closer and closer to \(2\), the function \(f(x)\) approaches a height of \(2\).

Key Concepts

Limits and Continuity

Understanding limits is fundamental when diving into calculus, as they are used to define important concepts such as continuity, derivatives, and integrals. Simply put, the limit of a function at a certain point refers to the value that the function 'approaches' as its input (or 'x' value) gets closer and closer to that point. But what makes a function continuous? A function is considered continuous at a point if the limit as you approach the point from both the left and the right exists and is equal to the function's value at that point.

For the given exercise, the challenge is to find the limit of a function as x approaches 2. If the function was continuous at x=2, we would expect the limit and the function's actual value at x=2 to be the same. However, in this case, we have a piecewise function where the value at x=2 is explicitly given as 0 while the limit, as computed, is 2. This tells us that there is a discontinuity at x=2, which is characteristic of piecewise functions with different behaviors around points of interest.

Piecewise Functions

Piecewise functions are like mathematical patchwork quilts; they are constructed from multiple functions, each applying to different intervals or 'pieces' of the domain. Each 'piece' of the function behaves differently depending on the input value. The common notation for piecewise functions is to list the function rules alongside the conditions under which they apply.

The given exercise presents a piecewise function with two pieces. For all values of x except 2, the function follows the rule described by the linear equation, y = 4 - x. However, when x is exactly 2, the function takes a detour and outputs a value of 0. This abrupt change creates what's called a 'discontinuity' at x = 2, leading to interesting limits behavior as x approaches this crucial point.

Approaching a Point

The concept of 'approaching a point' is central to understanding limits in calculus. When we say that x approaches a certain value, it means that x is getting infinitely close to that value but does not necessarily have to reach it. It's a bit like getting closer and closer to the edge of a cliff without actually stepping off. With limits, we are interested in the trend or the journey of the function's output, not necessarily where it stands at the precise point.

In our exercise, the function approaches a different value (namely, 2) than what it is actually defined at (which is 0) when x = 2. This has to do with the behavior of the parts of the piecewise function around the point x=2 and offers rich insight into how functions can behave near certain points, even if they aren't defined or continuous at those points.