Problem 58 - Thomas Calculus

Question

Let $$h(x)=\left\\{\begin{array}{ll}{x^{2},} & {x<2} \\ {3,} & {x=2} \\ {2,} & {x>2}\end{array}\right.$$ $$ \begin{array}{l}{\text { Show that }} \\ {\text { a. } \lim _{x \rightarrow 2} h(x) \neq 4} \\ {\text { b. } \lim _{x \rightarrow 2} h(x) \neq 3} \\\ {\text { c. } \lim _{x \rightarrow 2} h(x) \neq 2}\end{array} $$

Step-by-Step Solution

Answer

The limit of \(h(x)\) as \(x\) approaches 2 does not exist; therefore, it is not equal to 4, 3, or 2.

Step 1: Understanding the Limit Concept

To determine \(\lim_{x \to 2} h(x)\), we need to evaluate the limits from both the left and right of 2. A limit exists if both left-hand and right-hand limits are equal.

Step 2: Evaluate Left-Hand Limit

To find \(\lim_{x \to 2^-} h(x)\), observe the piecewise function as \(x<2\). Here, \(h(x) = x^2\). Calculate the limit as: \(\lim_{x \to 2^-} x^2 = 2^2 = 4\).

Step 3: Evaluate Right-Hand Limit

To find \(\lim_{x \to 2^+} h(x)\), observe the piecewise function as \(x>2\). Here, \(h(x) = 2\). Calculate the limit as: \(\lim_{x \to 2^+} 2 = 2\).

Step 4: Compare Left and Right Limits

The left-hand limit is 4, and the right-hand limit is 2. Since \(\lim_{x \to 2^-} h(x) eq \lim_{x \to 2^+} h(x)\), the limit \(\lim_{x \to 2} h(x)\) does not exist.

Step 5: Addressing Subparts

Since the limit does not exist, it cannot be equal to 4, 3, or 2. Thus, \(\lim_{x \to 2} h(x) eq 4\), \(\lim_{x \to 2} h(x) eq 3\), and \(\lim_{x \to 2} h(x) eq 2\) are all true.

Key Concepts

Piecewise functions

Piecewise functions are fascinating in the world of calculus. They are defined by different expressions depending on the input value. This can make them tricky but also very practical for representing real-world scenarios where conditions change over an interval. Imagine programming a set of traffic lights: at different times, different lights are active. A piecewise function can perfectly model this!

In mathematics, a piecewise function like \( h(x) \) is composed of multiple sub-functions. Each applies to a particular interval of the domain. For instance in \( h(x) \), the function returns \( x^2 \) for \( x < 2 \), the constant 3 at \( x = 2 \), and 2 for \( x > 2 \).

This form is invaluable in solving problems because it allows examination of a function's behavior at specific points or intervals. By breaking it down into manageable parts, we can analyze the function in much greater detail without losing sight of its overall operation.

One-sided limits

Understanding one-sided limits is crucial when dealing with piecewise functions. A one-sided limit examines the behavior of a function as it approaches a certain point from only one side: left or right. This is especially useful when the expression changes at specific points like with \( h(x) \).

We denote the left-hand limit as \( \lim_{x \to a^-} f(x) \), which means approaching \( x \) from values less than \( a \). Similarly, the right-hand limit \( \lim_{x \to a^+} f(x) \) examines the approach from values greater than \( a \).

In the context of \( h(x) \), to find \( \lim_{x \to 2^-} h(x) \), we consider the portion where \( x < 2 \). Here, \( h(x) = x^2 \), so we calculate \( \lim_{x \to 2^-} x^2 = 4 \).

Similarly, the right-hand limit looks at \( x > 2 \), where \( h(x) = 2 \), resulting in \( \lim_{x \to 2^+} h(x) = 2 \).

One-sided limits are vital tools. They allow us to handle functions that switch their expressions at certain values. Focusing on one direction at a time simplifies analyses of sections of a piecewise function.

Nonexistent limits

Sometimes, when a function's behavior doesn't align from both directions at a particular point, we encounter nonexistent limits. A limit is considered nonexistent if the left-hand limit and right-hand limit are not the same. This can occur at points where a piecewise function has different behaviors on either side of the point.

In the function \( h(x) \), the left-hand limit at \( x = 2 \) is 4, while the right-hand limit is 2. These are not equal, indicating that \( \lim_{x \to 2} h(x) \) does not exist. When limits don't match up, the function doesn't "settle down" to a single value from both sides.

Nonexistent limits play a critical role in calculus and analysis. They help in understanding discontinuities, which are points where the function fails to be seamless or smooth. This concept is essential for thoroughly examining and predicting a function's overall behavior in different mathematical and real-world applications.