Question

Piecewise Functions Let \(f(x)=\left\\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\\ {1,} & {\text { if } x \text { is irrational }}\end{array}\right.\) and \(g(x)=\left\\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\\ {x,} & {\text { if } x \text { is irrational }}\end{array}\right.\) Find (if possible) \(\lim _{x \rightarrow 0} f(x)\) and \(\lim _{x \rightarrow 0} g(x)\)

Short answer

The limit as x approaches 0 for \(f(x)\) does not exist, and the limit for \(g(x)\) is 0.

Step-by-step solution

Determine the behavior of f(x) around 0

At x=0, which is a rational number, f(x) equals to 0. However, for x approaching 0 but not equal to 0, x can either be rational or irrational. When x is a rational number, f(x) is 0. When x is an irrational number, f(x) is 1. Since the values of f(x) are differing depending on whether x is rational or irrational, we cannot find a single real number that f(x) approaches as x tends to 0. Therefore, the limit does not exist.

Determine the behavior of g(x) around 0

At x=0, which is a rational number, g(x) equals to 0. However, for x approaching 0 but not equal to 0, it can either be rational or irrational number. When x is a rational number, g(x) is 0. When x is an irrational number, g(x) is equal to x. Since as x gets closer to 0, the value of x also gets closer to 0. While the value alternates between 0 and a number very close to 0, we can still say that g(x) approaches 0. Thus, the limit of g(x) as x tends to 0 is 0.

Key concepts

Limit of a Function

The concept of the limit of a function is a fundamental idea in calculus, and it helps us understand how a function behaves as it approaches a certain point. When we say \[ \lim_{x \to a} f(x) = L \]we mean that as the input \(x\) gets closer and closer to \(a\), the output \(f(x)\) gets closer and closer to \(L\). For a limit to exist at a particular point, the left-hand and right-hand limits must both approach the same value \(L\).
In the case of the piecewise function \(f(x)\) defined in this exercise, because \(f(x)\) takes different values depending on whether \(x\) is rational or irrational, the behavior of the function fluctuates too much as \(x\) approaches 0. Thus, no single value \(L\) can satisfy the condition, and therefore, the limit does not exist.
Understanding limits is crucial as they play a vital role in defining the concept of continuity, derivatives, and integrals, which are key components of calculus.

Rational and Irrational Numbers

Rational and irrational numbers are the building blocks of the real number system. Rational numbers are numbers that can be expressed as the quotient of two integers, \(p/q\), where \(q\) is not zero. Examples include 1/2, 3, -7, and 0.

• They have finite or repeating decimal expansions, such as 0.5 or 0.333...
• Every integer is a rational number since it can be expressed as the quotient of itself and 1.

Irrational numbers, in contrast, cannot be expressed as a simple fraction. Their decimal expansions are non-repeating and non-terminating. Famous examples include \( \sqrt{2} \), \( \pi \), and \( e \).
The function \(f(x)\) in the exercise takes two different values depending on whether \(x\) is rational or irrational. This distinction is crucial to understanding the piecewise nature of these functions, especially when discussing limits or continuity.

Behavior of Functions Around a Point

When analyzing the behavior of a function around a point, it's important to observe how the function behaves not just at the point itself, but as it approaches from both sides. This is where left-hand limits \( \lim_{x \to a^-} \) and right-hand limits \( \lim_{x \to a^+} \) come into play.
In our example with \(f(x)\), as \(x\) approaches 0, whether \(x\) is rational or irrational causes the function to settle at different values. This inconsistency in behavior means that the function cannot settle on a single value as it approaches the point from both sides, leading the limit to be non-existent.
For \(g(x)\), the function also takes on different values based on the nature of \(x\), but as \(x\) gets closer to 0, \(g(x)\) tends to zero itself, indicating that the function has a well-defined behavior around 0.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It provides tools for analyzing how things change and is essential for many areas of science and engineering.

• The study of limits forms the foundation for entire calculus concepts such as derivatives and integrals.
• Understanding the behavior of functions, whether through limits or continuity, is essential for solving real-world problems.

In this particular problem, understanding calculus helps in determining how piecewise functions behave around certain points, particularly through limits. Despite \(f(x)\) not having a limit at 0, calculus allows us to formalize and understand why \(g(x)\) does. It synthesizes rational and irrational behavior into a framework that explains real-world dynamics.