Problem 26 - Calculus Early Transcendentals

Question

Prove the statement using the \(\varepsilon, \delta\) definition of limit. \(\lim _{x \rightarrow 0} x^{3}=0\)

Step-by-Step Solution

Answer

For every \(\varepsilon > 0\), set \(\delta = \varepsilon^{1/3}\) to satisfy \(|x^3| < \varepsilon\) when \(0 < |x| < \delta\).

Step 1: Understand the Definition

To prove that \(\lim_{x \to 0} x^3 = 0\) using the \(\varepsilon, \delta\) definition of limit, we must show that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x| < \delta\), then \(|x^3 - 0| < \varepsilon\).

Step 2: Setup the Inequality

We need to make the expression \(|x^3 - 0| < \varepsilon\). Simplifying this, we get \(|x^3| < \varepsilon\).

Step 3: Choose Delta

Consider \(|x|\) such that \(|x^3| = |x|^3 < \varepsilon\). To satisfy this inequality, we need \(|x| < \varepsilon^{1/3}\). Thus, set \(\delta = \varepsilon^{1/3}\).

Step 4: Verify the Condition

Ensure that our choice of \(\delta\) works. If \(0 < |x| < \delta = \varepsilon^{1/3}\), then \(|x|^3 < (\varepsilon^{1/3})^3 = \varepsilon\). Therefore, \(|x^3| < \varepsilon\), which satisfies the requirement.

Step 5: Conclusion

Since for every \(\varepsilon > 0\) we can find \(\delta = \varepsilon^{1/3} > 0\) such that if \(0 < |x| < \delta\), then \(|x^3 - 0| < \varepsilon\), we have proven that \(\lim_{x \to 0} x^3 = 0\) using the \(\varepsilon, \delta\) definition.

Key Concepts

Limits

In calculus, limits help us understand how functions behave as their input approaches a certain value. It's like looking at the road signs leading to a city to anticipate how the traffic will change as you get closer.
To determine a limit, we explore what happens when the input value gets infinitely close to a specific point. The \(x\) in our function approach a target, this can either be a number or infinity. Then, we observe the behavior of our function’s output.
In the exercise, we're asked to explore the behavior of \(x^3\) as \(x\) approaches zero. This translates to finding out what value \(x^3\) gets near when \(x\) is nearly zero. This foundational principle in calculus sets the stage for exploring continuity and derivative calculations.

Continuity

Continuity means looking at a function to see if it behaves nicely or smoothly without any disruptions, jumps, or breaks. Imagine a graph where you can draw it in one go, without lifting your pen.
For a function \(f(x)\) to be continuous at a point \(x = c\), three conditions should be satisfied:

• \(f(c)\) must be defined.
• The limit of \(f(x)\) as \(x\) approaches \(c\) should exist.
• The limit of \(f(x)\) as \(x\) approaches \(c\) must equal \(f(c)\).

The exercise focuses on the limit of \(x^3\) at \(x = 0\). We could say that \(x^3\) is continuous at this point since, indeed, \(0^3 = 0\), and all three conditions are met because the function behaves smoothly without breaking away when \(x\) tends to zero.

Calculus

Calculus is the mathematical study of change. It helps us model and understand how things move and grow in our world by examining small changes and their consequences.
Divided into differential and integral calculus, it first explores things like how the speed of a car at a specific moment changes (differentiation). Then, it measures accumulated quantities like the distance traveled over time (integration).
The epsilon-delta definition falls under differential calculus, focusing on limits. It gives us a precise way to confirm that a function's output is predictable and consistent as the input nears a specific value. In the exercise, proving the limit of \(x^3\) at zero helps us grasp the concept of stability in prediction about how \(x^3\) behaves for very small values of \(x\).

Proof Writing

Proof writing in mathematics is like crafting a compelling story or a rigorous argument that convinces others to believe in a certain truth. It's the art of logically showing why something is true through clear and structured reasoning.
When writing proofs using the epsilon-delta definition, like in our exercise, follow these key steps:

• Clearly state what you need to prove, such as a limit statement.
• Understand and outline the definition involving \(\varepsilon > 0\) and find a corresponding \(\delta > 0\).
• Solve for constraints on your variables, here it's translating \(|x^3| < \varepsilon\) into a form that's easier to handle.
• Choose a suitable \(\delta\), validate it satisfies the conditions, and conclude your proof.

This logical journey not only reinforces understanding of calculus concepts but also hones critical thinking and precise communication skills. Writing the proof for \(\lim _{x \rightarrow 0} x^{3}=0\) meticulously establishes confidence in the concept and builds a solid foundation for future mathematical explorations.