Question

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 0} \sqrt[3]{x} $$

Short answer

The limit \(L\) of \(\sqrt[3]{x}\) as \(x\) approaches 0 is 0. The \(\varepsilon-\delta\) proof, for every \(\varepsilon > 0\), there exists a \(\delta = \varepsilon^3\) such that if \(0 < |x - 0| < \delta = \varepsilon^3\), then \(|\sqrt[3]{x} - 0| < \varepsilon\).

Step-by-step solution

Find the Limit

First, this problem requires to find the limit of \(\sqrt[3]{x}\) when \(x\) approaches 0, which is actually 0. So, \(L = 0\). The limit equation can be written as: \(\lim_{{x \rightarrow 0}} \sqrt[3]{x} = 0\).

Use the \(\varepsilon-\delta\) Definition

Second for the \(\varepsilon-\delta\) definition, one says that \(|f(x) - L| < \varepsilon\) whenever \(0 < |x - c| < \delta\), where \(c\) is the point \(x\) is approaching, in this case 0. So the equation to prove is \(|\sqrt[3]{x} - 0| < \varepsilon\) whenever \(0 < |x - 0| < \delta\). This simplifies to \(|\sqrt[3]{x}| < \varepsilon\) whenever \(0 < |x| < \delta\), or, as we can say that the abs value of a positive real number is the number itself, \(\sqrt[3]{x} < \varepsilon\) whenever \(0 < x < \delta\).

Find \(\delta\) according to \(\varepsilon\)

In sequence, we need to express \(\delta\) in terms of \(\varepsilon\) fulfilling the condition that \(\sqrt[3]{x} < \varepsilon\) whenever \(0 < x < \delta\). It can be observed that if we cube both sides, the inequality will remain the same since both sides are positive and we get \(x < \varepsilon^3\). Here it can be determined, \(\delta = \varepsilon^3\). Put \(\delta\) in the second step equation, we can get whenever \(0 < x < \varepsilon^3\), \(\sqrt[3]{x}< \varepsilon\). So the proof has been completed.

Key concepts

Limit of a Function

To understand the concept of the limit of a function, imagine you have a function that describes the behavior of something, like the path of a moving car. The limit helps you figure out where the car is headed as time goes on. Mathematically, when we say \(\lim_{{x \rightarrow c}} f(x) = L\), it means that as \(x\) gets closer and closer to \(c\), the function \(f(x)\) approaches the value \(L\).
This concept is very helpful when dealing with situations where the function doesn't necessarily "reach" the value \(L\) at the point \(c\). It only matters what value \(f(x)\) gets close to as \(x\) gets closer to but does not necessarily equal \(c\).
Understanding limits introduces you to continuous functions, where a small change in \(x\) produces a tiny change in \(f(x)\). Breaking it down, the limit resembles the predictive power of mathematics, explaining how things behave under certain conditions.

Cubic Root

The cubic root of a number, denoted as \(\sqrt[3]{x}\), is the value that, when multiplied by itself three times, gives the original number \(x\). For example, the cubic root of 27 is 3 because \(3 \times 3 \times 3 = 27\).
While square roots are more commonly encountered, cubic roots provide insights into three-dimensional scaling and algebraic transformations where we are interested in the root in terms of multiplication.
In the problem we're working with, we focused on finding the limit of the cubic root as \(x\) approaches zero. Keep in mind, the concept may seem daunting, but grasping how roots and powers function will be invaluable as you delve deeper into calculus and higher-level math.

Mathematical Proof

Mathematical proof is all about establishing the truth of a statement beyond doubt by following logical reasoning. It ensures mathematicians are not just guessing, but are certain about the truth of specific conclusions.
To construct a proof, you start with known facts or assumptions, proceed through logical steps, and end with a conclusion that is undeniably true given those initial premises.
This rigorous process helps explain why mathematical facts are reliable and universally accepted. In the given exercise, our proof lies in proving the limit using the epsilon-delta definition, which is like providing a watertight logical argument that guarantees the truth of the limit statement.

Delta-Epsilon Proof

The Delta-Epsilon Proof is a technique in calculus used to rigorously prove limits. It provides a formal way to say, "No matter how small of a range I set for \(f(x)\), I can set \(x\) close enough to \(c\) such that \(f(x)\) stays within this range."
For this proof, we first identify \(\varepsilon\), a small positive number that represents the distance from \(L\) (the limit). Then, we find \(\delta\), another small positive number, ensuring that as long as \(x\) is within \(\delta\) of \(c\), \(f(x)\) remains within \(\varepsilon\) of \(L\).

• We showcase that with our chosen \(\delta\), the inequality \(|f(x) - L| < \varepsilon\) holds.
• In our specific case, we found \(\delta = \varepsilon^3\).

So, the Delta-Epsilon Proof is an elegant and powerful way to demonstrate limits, providing assurance that our math stays precise.