Question

Find the limit \(L\). Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L\). $$ \lim _{x \rightarrow 6} 3 $$

Short answer

The limit \(\lim _{x \rightarrow 6} 3 = L = 3\). For any arbitrary \(\varepsilon > 0\), we can choose any \(\delta >0\) (even the smallest positive number), and the limit statement will hold true. Therefore, the \(\varepsilon-\delta\) definition verifies our conclusion that the limit equals 3.

Step-by-step solution

Determine value of L

For the limit \(\lim _{x \rightarrow a} c\) with c as a constant, the limit is always the constant itself. Hence, L would be 3.

Explain the \(\varepsilon-\delta\) definition

The \(\varepsilon-\delta\) definition is a formal definition of a limit. It states that for every real \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if the absolute value of the difference between the variable x and the limit point a is less than \(\delta\) (excluding the case where x = a), then the absolute value of the difference between the function f(x) and L is less than \(\varepsilon\). That is, \(0 < |x - a| < \(\delta\) implies |f(x) - L| < \(\varepsilon\).

Use \(\varepsilon-\delta\) to prove limit

Here, we're trying to prove that \(\lim _{x \rightarrow 6} 3 = L = 3\). We'll take an arbitrary \(\varepsilon > 0\). Our aim is to find a \(\delta\) such that whenever \(0 < |x - 6| < \delta\), it follows that |3 - 3| < \(\varepsilon\). As |3 - 3| = 0 (which is less than any positive number), it is clear that we can make |3 - 3| < \(\varepsilon\) for any \(\varepsilon > 0\). Therefore, \(\lim _{x \rightarrow 6} 3 = 3\) holds true for any \(\delta\) (even if you take \(\delta\) to be the smallest positive number).

Key concepts

Limits in Calculus

The concept of a limit is a fundamental building block in calculus. It helps us in understanding the behavior of functions as an input approaches a particular value. For instance, when we say \(\lim _{x \rightarrow a} f(x) = L\), we're essentially looking at what the function value \(f(x)\) gets closer to, as \(x\) gets closer and closer to \(a\). Limits are crucial in defining instantaneous rates of change, which lead to the concept of derivatives, and in understanding the behavior of functions that aren't easily evaluated at certain points.

Learning about limits involves familiarizing oneself with various limit laws and knowing when and how to apply these laws to find the value of a limit. It also involves understanding when limits do not exist, such as when a function approaches infinity or when a function oscillates without approaching any particular value.

Proof of Limits

The \(\varepsilon-\delta\) definition of a limit is a precise mathematical notion that helps in proving the value of a limit. It sets a rigorous framework for establishing that a function \(f(x)\) approaches the limit \(L\) as \(x\) approaches \(a\). The proof involves two stages: first showing the existence of the limit, and then demonstrating that the function indeed approaches this limit within an acceptable margin of error, \(\varepsilon\), for every \(\delta\) neighborhood around \(a\).

This concept is critical for a deeper understanding of calculus as it removes ambiguity and lays a foundation for more advanced topics. It essentially asks us to find such a \(\delta\), no matter how small, that ensures \(f(x)\) stays within \(\varepsilon\) of \(L\) when \(x\) is within \(\delta\) of \(a\), excluding the exact point \(x = a\). The \(\varepsilon-\delta\) proof is the gold standard for limit verification in calculus.

Calculus Concepts

Calculus is composed of a variety of concepts that deal with continuous change. These include derivatives, which represent the rate of change of a function, and integrals, which measure area under a curve or accumulate quantities. Other concepts include continuity, a property of functions that are unbroken and intuitive, and infinite sequences and series, which allow us to sum an endless list of numbers in a meaningful way.

Understanding calculus requires a solid grasp of limits, as they are intertwined with nearly every other concept in calculus. Without the foundational knowledge of limits and their properties, delving deeper into derivatives and integrals can be challenging. Calculus is not just a collection of techniques, but also a way of thinking about the world, modeling dynamic systems, and solving complex problems across various disciplines.