Question

Proof of the Squeeze Theorem Assume the functions \(f, g,\) and \(h\) satisfy the inequality \(f(x) \leq g(x) \leq h(x),\) for all \(x\) near \(a\) except possibly at \(a\). Prove that if \(\lim _{x \rightarrow a^{+}} f(x)=\lim _{x \rightarrow a} h(x)=L\) then \(\lim _{x \rightarrow a} g(x)=L\)

Short answer

Short Answer: To prove that \(\lim_{x \rightarrow a} g(x) = L\), given that \(f(x) \leq g(x) \leq h(x)\), and \(\lim_{x \rightarrow a^{+}} f(x) = \lim_{x \rightarrow a} h(x) = L\), we use the definition of limits and show that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x-a| < \delta\), then \(|g(x) - L| < \epsilon\). We combine the inequalities involving \(f(x)\), \(g(x)\), and \(h(x)\), and show that \(|g(x) - L| < \epsilon\), concluding that \(\lim_{x \rightarrow a} g(x) = L\).

Step-by-step solution

State what needs to be proven

We want to prove that if \(f(x) \leq g(x) \leq h(x)\) for all \(x\) near \(a\), except possibly at \(a\), and \(\lim_{x \rightarrow a^{+}} f(x) = \lim_{x \rightarrow a} h(x) = L\), then \(\lim_{x \rightarrow a} g(x) = L\).

Use the definition of limits

The definition of a limit states that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x-a| < \delta\), then \(|g(x) - L| < \epsilon\).

Apply the given conditions

We know that \(f(x) \leq g(x) \leq h(x)\) and \(\lim_{x \rightarrow a^{+}} f(x) = \lim_{x \rightarrow a} h(x) = L\). So, for any \(\epsilon > 0\), there exist \(\delta_1, \delta_2 > 0\) such that: 1. If \(0 < |x - a| < \delta_1\), then \(|f(x) - L| < \epsilon\). 2. If \(0 < |x - a| < \delta_2\), then \(|h(x) - L| < \epsilon\).

Combine the inequalities

Let \(\delta = \min(\delta_1, \delta_2)\). Then, for \(0 < |x - a| < \delta\), we have both \(|f(x) - L| < \epsilon\) and \(|h(x) - L| < \epsilon\). Since \(f(x) \leq g(x) \leq h(x)\), we also have: 1. \(L - \epsilon < f(x) \leq g(x)\). 2. \(g(x) \leq h(x) < L + \epsilon\).

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

Combining the inequalities from Step 4, we have: \(L - \epsilon < f(x) \leq g(x) \leq h(x) < L + \epsilon\). Thus, \(L - \epsilon < g(x) < L + \epsilon\), which implies \(|g(x) - L| < \epsilon\). Since this is true for any \(\epsilon > 0\), we can conclude that \(\lim_{x \rightarrow a} g(x) = L\).

Key concepts

Calculus

Calculus is a branch of mathematics fundamentally concerned with change — more specifically, how things change and at what rate they do so. It's divided into two sub-fields: differential calculus and integral calculus. Differential calculus deals with the concept of the derivative, which represents the rate of change of a function with respect to a variable, while integral calculus is concerned with the concept of the integral, which represents the accumulation of quantities.

Understanding the behavior of functions as they approach certain values is an essential part of calculus and is facilitated by studying limits. The study of limits helps us to define and understand continuity, derivatives, and integrals. They give us a rigorous way to discuss and calculate the 'approach' toward a certain value, even if the function never actually reaches that value.

For instance, in exploring the effect of change around a point, the Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, comes into play. This theorem allows us to find the limit of a function via the limits of two other functions that 'squeeze' it from above and below. In essence, if a function is always greater than or equal to one function and less than or equal to another, and if these two functions approach the same limit at a point, so must the function being squeezed.

This concept is deeply connected with real-world scenarios, such as using physics to predict the limit of an object's position, as its speed approaches a constant value when the object is caught between two converging forces.

Limits

In calculus, the concept of limits can be quite abstract but it forms the building block for more complex ideas such as derivatives and integrals. A limit basically answers the question, 'What value does a function approach as the input approaches a certain value?' This notion is precisely what underlies both the challenge and utility of calculus.

For example, suppose you're examining the temperature change as you move toward the edge of a metal bar that's being heated at one end. The limit of the temperature function as you move towards that heated end tells you the temperature you expect to find, without needing to actually touch the end and potentially burn yourself.

The exercise at hand uses the limit concept to prove the Squeeze Theorem. It embodies the fundamental aspect of limits: the behavior of functions as they get arbitrarily close to a specific point. The solution to the problem demonstrates how we can confidently determine the limit of a function at a point even if the function doesn't actually reach that value at the point, provided certain conditions are met. This proof technique uses the fact that, by understanding the behavior of limits for functions that bound another, one can deduce the behavior of the function in the middle.

Proof Techniques

Proof techniques are logical arguments that establish the truth of mathematical statements and are fundamental in making compelling and reliable mathematical conclusions. They are the foundation of mathematics because they provide the means to assert with certainty that certain propositions are indeed true.

There are diverse methods to prove mathematical statements, including direct proofs, proof by contradiction, proof by induction, and more. The step by step solution demonstrated uses a direct proof, a method where the conclusion is reached by a straightforward application of known facts and definitions.

In proving the Squeeze Theorem, the direct proof begins with the definition of limits and what is known about the functions involved. From there, it logically deduces, step by step, that the limit of the function within the 'squeeze' is the same as the limits on either side. The exercise progresses from known properties of the functions and uses the logical steps to establish the desired conclusion.

Such proofs not only help in understanding the material on a deeper level but also train students to develop rigorous logical thinking, which is applicable outside the realm of mathematics in various forms of problem-solving and decision-making activities.