Problem 61 - Calculus: Early Transcendentals

Question

If \( f(x) = \left\\{ \begin{array}{ll} x^2 & \mbox{if \) x \( is rational}\\\ 0 & \mbox{if \) x \( is irrational} \end{array} \right.\) prove that \( \displaystyle \lim_{x \to 0}f(x) = 0 \).

Step-by-Step Solution

Answer

The limit is 0 as both rational \( x^2 \to 0 \) and irrational \( f(x) \to 0 \) near 0.

Step 1: Understand the Function

The function \( f(x) \) is defined in a piecewise manner, where \( f(x) = x^2 \) if \( x \) is rational, and \( f(x) = 0 \) if \( x \) is irrational.

Step 2: Conceptualize Limit Definition

The exercise requires us to prove that \( \lim_{x \to 0}f(x) = 0 \), meaning that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - 0| < \delta \), then \( |f(x) - 0| < \epsilon \).

Step 3: Apply Limit Definition to Rational x

For rational \( x \), \( f(x) = x^2 \). So, we need to ensure \( |x^2| < \epsilon \). Since \( x^2 = |x|^2 \), choose \( \delta = \sqrt{\epsilon} \) such that if \( 0 < |x| < \delta \), then \( |x^2| < \epsilon \).

Step 4: Apply Limit Definition to Irrational x

For irrational \( x \), \( f(x) = 0 \). We need \( |0 - 0| = 0 < \epsilon \), which is always true regardless of the choice of \( \delta \), ensuring \( |f(x) - 0| < \epsilon \).

Step 5: Combine Rational and Irrational Cases

Since both cases meet the \( \epsilon \)-\( \delta \) criterion for \( x \to 0 \), we conclude \( \lim_{x \to 0}f(x) = 0 \), as both rational and irrational sequences \( x \to 0 \) result in \( f(x) \to 0 \).

Key Concepts

Piecewise Functions

Piecewise functions are special kinds of functions where different parts of the function apply to different parts of their domain. This means the function behaves differently based on certain conditions. In this exercise, the function \( f(x) \) is defined in a piecewise manner, depending on whether \( x \) is rational or irrational.
\( f(x) = x^2 \) if \( x \) is a rational number. For any irrational \( x \), \( f(x) = 0 \).
This kind of setup is useful in modeling situations where a function needs to adapt its behavior based on varying conditions. Understanding a piecewise function involves examining each "piece" individually to comprehend the overall behavior of the function.

Epsilon-Delta Definition

The epsilon-delta definition of a limit is a precise way to describe what it means for a function to approach a certain value as the input gets closer to some point. This definition is critical in calculus and provides the foundation for evaluating limits.
According to the epsilon-delta definition: for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \epsilon \), where \( c \) is the point x approaches and \( L \) is the value \( f(x) \) approaches.
In this problem, we apply the epsilon-delta definition to demonstrate that as \( x \to 0 \), both rational and irrational values of \( x \) move \( f(x) \) closer to 0, thus satisfying the condition for the limit.

Rational and Irrational Numbers

Understanding rational and irrational numbers is crucial to grasping the behavior of piecewise functions such as the one in this exercise. Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).
Irrational numbers, on the other hand, cannot be expressed as such fractions. They have non-repeating, non-terminating decimal expansions. Examples of irrational numbers include \( \pi \), \( e \), and \( \sqrt{2} \).
The function \( f(x) \) in this exercise behaves differently for rational and irrational numbers, which corresponds to the specific values these numbers can take, impacting the function's outcome near limits.

Continuity

Continuity is a fundamental property in calculus that describes whether a function behaves in an uninterrupted manner as you move along its graph. A function is continuous at a point if the following three conditions are fulfilled:

• The function is defined at the point.
• The limit of the function exists as x approaches that point.
• The limit of the function equals the function’s value at that point.

For \( f(x) \) in this exercise, which changes its behavior based on whether \( x \) is rational or irrational, continuity around the point \( x = 0 \) may seem challenging. However, by demonstrating that both the rational and irrational approaches of \( x \to 0 \) result in \( f(x) \to 0 \), we establish continuity at this point according to the formal mathematical conditions.