Question

True or False? In Exercises \(115-120\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \lim _{x \rightarrow 2} f(x)=3, \text { where } f(x)=\left\\{\begin{array}{ll}{3,} & {x \leq 2} \\ {0,} & {x>2}\end{array}\right. $$

Short answer

The statement is false. The limit at \(x = 2\) does not exist because the function approaches different values from the left side and the right side of \(x = 2\).

Step-by-step solution

Understand the function

In the given function, \(f(x)\), the value is divided into two sections. It is 3 whenever \(x\) is less than or equal to 2, and 0 whenever \(x\) is greater than 2.

Analyze the limit of the function as \(x\) approaches 2

We need to figure out what value \(f(x)\) approaches as \(x\) gets closer and closer to 2. Since we have two different definitions of \(f(x)\), depending if \(x\) is less than or equal to 2, or \(x\) is greater than 2, we need to consider them separately. When \(x\) approaches 2 from the left side (that means when \(x\) is slightly less than 2), according to the function definition, \(f(x)\) will approach 3. On the other hand, when \(x\) approaches 2 from the right side (\(x\) is slightly more than 2), \(f(x)\) will approach 0.

Conclude the truthfulness of the statement

The given statement is true only if the limit is equal to 3. But we figured out in the previous step that though the function approaches 3 from the left side of 2, it approaches 0 from the right side of 2. Hence, the limit does not exist at \(x = 2\). Therefore, the given statement that \(\lim _{x \rightarrow 2} f(x)=3\) is false.

Key concepts

Piecewise Functions

A piecewise function is a type of function that is defined by different expressions based on different intervals of the input variable, usually denoted as \(x\). This means the function can change its formula depending on the value of \(x\). In our example, the piecewise function is defined as follows:

• If \(x \leq 2\), then \(f(x) = 3\).
• If \(x > 2\), then \(f(x) = 0\).

Piecewise functions are essential for representing real-world situations where a condition may change, such as a cost model that depends on quantity. They are also vital in understanding limits because the behavior of a function can change at specific points.

Left-hand Limit

The left-hand limit (also known as the limit from the left) examines the behavior of a function as the variable \(x\) approaches a particular point from the left side, or from smaller values. In simpler terms, it looks at what happens to a function as \(x\) gets closer and closer to some number from the lower side.

For the given function, as \(x\) approaches 2 from smaller values (that is, values less than 2), the function value \(f(x)\) approaches 3. This is because the section of the piecewise function used for values when \(x \leq 2\) always outputs 3. Therefore, the left-hand limit of this function as \(x\) approaches 2 is 3.

Understanding left-hand limits is crucial when dealing with piecewise functions, especially when analyzing discontinuities.

Right-hand Limit

The right-hand limit checks the behavior of a function as the variable \(x\) approaches a particular point from the right side, or from larger values. When we talk about approaching from the right, we mean that \(x\) is slightly more than a particular value.

In this specific example, as \(x\) approaches 2 from larger values (or \(x > 2\)), the function \(f(x)\) approaches 0 which is defined by the piecewise function for values when \(x > 2\). Hence, the right-hand limit of the function as \(x\) approaches 2 is 0.

Right-hand limits help us comprehend the function's behavior on the other side of a given point. They are pivotal when determining whether a function is continuous or has a jump at a specific point.

Discontinuity in Functions

A discontinuity occurs in functions where there is a sudden jump or break. Such a spot is where a function does not behave consistently across its domain. When the left-hand limit and right-hand limit at a certain point are not equal, a discontinuity occurs at that point.

For the problem at hand, the left-hand limit at \(x = 2\) is 3, while the right-hand limit is 0. Since these two limits are not the same, it indicates a discontinuity at \(x = 2\). The function suddenly jumps from a value of 3 to 0 as \(x\) crosses over 2.

Understanding discontinuities is critical because they tell us where functions might behave unexpectedly, and they play an essential role in calculus, particularly when evaluating limits and integrals.