Question

Limit proof Suppose \(f\) is defined for all \(x\) near \(a\), except possibly at \(a\). Assume for any integer \(N>0,\) there is another integer \(M>0\) such that \(|f(x)-L|<1 / N\) whenever \(|x-a|<1 / M\) Prove that \(\lim _{r \rightarrow a} f(x)=L\) using the precise definition of a limit.

Short answer

In this exercise, we have successfully shown that the given information on integers \(N\) and \(M\) is sufficient to prove the limit \(\lim_{x\rightarrow a}f(x) = L\) exists. The key steps were to choose an appropriate \(N\) and the corresponding \(M\), and define a value for \(\delta\) before showing that the definition of the limit holds for the given information.

Step-by-step solution

Assume an arbitrary \(\epsilon > 0\)

Let \(\epsilon > 0\). Our goal is to find a \(\delta > 0\) such that for every \(x\) with \(0 < |x-a| < \delta\), \(|f(x) - L| < \epsilon\).

Choose an appropriate \(N\) and \(M\)

Choose an integer \(N\) such that \(\frac{1}{N} < \epsilon\). Since \(N > 0\), there exists an integer \(M\) corresponding to this \(N\) such that \(|f(x) - L| < \frac{1}{N}\) whenever \(|x-a| < \frac{1}{M}\).

Define the value of \(\delta\)

Define \(\delta = \frac{1}{M}\). Now, for any \(x\) with \(0 < |x-a| < \delta = \frac{1}{M}\), we know that \(|f(x) - L| < \frac{1}{N}\).

Show that \(|f(x) - L| < \epsilon\)

Since we chose \(N\) such that \(\frac{1}{N} < \epsilon\), combining this inequality with the previous step where we have \(|f(x) - L| < \frac{1}{N}\), we get: \(|f(x) - L| < \frac{1}{N} < \epsilon\) This shows that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that \(0 < |x-a| < \delta \Rightarrow |f(x)-L| < \epsilon\), proving that \(\lim_{x\rightarrow a}f(x) = L\).

Key concepts

Precise Definition of a Limit

In calculus, the precise definition of a limit is foundational for understanding how functions behave as inputs approach a specific value. This definition is crucial because it shifts the idea of limits from a numerical estimation to a mathematical statement that can be proved rigorously. Generally, the definition states: "We say that the limit of function \(f(x)\) as \(x\) approaches \(a\) is \(L\), and write \(\lim_{x\to a} f(x) = L\), if for every number \(\epsilon > 0\), there exists a number \(\delta > 0\) such that whenever \(0 < |x - a| < \delta\), it follows that \(|f(x) - L| < \epsilon\)."
This means we can make \(f(x)\) infinitely close to \(L\) by choosing \(x\) values sufficiently near \(a\) but not equal to \(a\). The \(\epsilon \) represents any small positive number, indicating how close \(f(x)\) should be to \(L\), and \(\delta\) tells us how close \(x\) needs to be to \(a\) to achieve this. Through this precise language, we avoid ambiguity and reinforce the precision needed in calculus.

Epsilon-Delta Proof

An epsilon-delta proof is an essential part of proving that a limit exists according to its precise definition. This proof formalism involves two parts: choosing a suitable \(\epsilon > 0\) and then finding a corresponding \(\delta > 0\) to meet the limit condition. Using the exercise, here's how you conduct an epsilon-delta proof:

• Choosing \(\epsilon\): Start by selecting any positive number \(\epsilon\) representing how close \(f(x)\) must be to \(L\).
• Finding \(\delta\): Identify \(\delta > 0\) for which \(f(x)\) stays within \(\epsilon\) of \(L\) whenever \(x\) is within \(\delta\) distance from \(a\). This might involve adjusting constants or variables like \(N\) and \(M\) in the exercise's settings.
• Verification: Demonstrate the chosen \(\delta\) achieves the condition \(0 < |x-a| < \delta\) leading to \(|f(x) - L| < \epsilon\).

The objective is to confirm that no matter how small we make \(\epsilon\), we can always find a \(\delta\) enabling \(f(x)\) to approach \(L\) as \(x\) approaches \(a\). Successfully establishing this relationship validates the existence of the limit as defined.

Limit of a Function

The limit of a function is a key concept in calculus, depicting the value that a function \(f(x)\) approaches as the input \(x\) approaches a certain point \(a\). Understanding this helps in predicting function behavior near points that might be undefined or difficult to assess otherwise. Limits are fundamental because they form the basis for defining derivatives and integrals.
In practice, finding a limit involves determining what value \(f(x)\) gets close to as \(x\) nears \(a\). The mathematical rigor through concepts like the epsilon-delta definition ensures the result's accuracy, surpassing trial and error methods. This rigorous approach allows us to:

• Conclude the function's behavior near discontinuities or points of interest.
• Engage in more advanced calculus topics, like continuity, differentiability, and the fundamentally related notion of the derivative, which relies on limits.
• Predict outcomes in real-world scenarios, such as calculating rates of change.

In summary, limits provide a way to precisely manage and understand functions' behaviors at crucial points, which bolsters our mathematical toolkit in analysis and application.