Problem 54 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Given three functions, \(f(x), g(x),\) and \(h(x)\), with the inequality \(f(x) \leq g(x) \leq h(x)\) for all values of \(x\) near \(a\), and \(\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} h(x) = L\), prove that \(\lim_{x \rightarrow a} g(x) = L\). Answer: By applying the Squeeze Theorem, since \(\lim_{x \rightarrow a} f(x) = L\) and \(\lim_{x \rightarrow a} h(x) = L\), then \(\lim_{x \rightarrow a} g(x) = L\).

Step 1: Recall the Squeeze Theorem

The squeeze theorem states that if \(f(x) \leq g(x) \leq h(x)\) for all \(x\) near \(a\) and \(\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} h(x) = L\), then \(\lim_{x \rightarrow a} g(x) = L\).

Step 2: Apply the Squeeze Theorem

We're given the conditions of the squeeze theorem, so we can directly apply it to the functions \(f(x), g(x),\) and \(h(x)\) to conclude that since \(\lim_{x \rightarrow a} f(x) = L\) and \(\lim_{x \rightarrow a} h(x) = L\), therefore \(\lim_{x \rightarrow a} g(x) = L\). So, we have proved that if \(\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} h(x)=L\) then \(\lim _{x \rightarrow a} g(x)=L\).

Key Concepts

Squeeze Theorem in Calculus

The Squeeze Theorem, also known as the Sandwich Theorem, is a fundamental concept in calculus that allows us to find the limit of a function when direct substitution or algebraic simplification isn't possible. The theorem's beauty lies in its simplicity and power: if you have three functions—let's call them 'f', 'g', and 'h'—and 'f' and 'h' tightly 'squeeze' 'g' such that at a certain point, 'a', all functions approach the same value, 'L', then 'g' is forced to also approach 'L' as its limit at that point.

Visualize three lines on a graph: 'f' is the lower line, 'h' is the upper line, and 'g' is sandwiched in between. As these lines zoom in towards the point 'a', 'f' and 'h' converge to a single value. Consequently, 'g' has no option but to follow suit, thereby revealing its limit through the pressure exerted by 'f' and 'h'. This principle comes in handy when 'g' is too complex to evaluate using standard limit-finding strategies.

One can think of it like a car in a one-lane tunnel with 'f' and 'h' being the tunnel walls. If the walls converge on a single point, the car ('g') must necessarily pass through this same point.

Limit of a Function

When we talk about the limit of a function in calculus, we're essentially discussing the behavior or value that a function approaches as its input gets closer to some number. It's a predictive tool, allowing us to forecast a function's trajectory without needing the function to be defined exactly at that point. For example, you might have a function 'g' that doesn't exist at 'x = a', but as you look at the values of 'g' when 'x' is incredibly close to 'a', both from the left and the right, you see that it appears to lean towards a certain number, 'L'.

The most exciting aspect is that limits help us understand continuity and evaluate functions at points where they're not explicitly defined. Imagine driving towards a bridge and just before you get there, it disappears! But if signs (values of 'g' for 'x' near 'a') along the road keep mentioning the bridge, you can confidently assume how to cross the river, even with the bridge (the function at 'x = a') out of sight. As long as the pattern is clear, you can deduce the presence of the bridge ('L')—that's what limits allow mathematicians to do.

Proving Limits Using Theorems

In calculus, to 'prove' the limit of a function means to mathematically demonstrate that as 'x' approaches a particular value, the function indeed gets closer to a specific limit. The Squeeze Theorem is just one tool in the mathematician's kit for such proofs. When a function is tricky and refuses to give away its limit through typical methods, like factorization or finding common denominators, theorems become our saviors.

Other than the Squeeze Theorem, there are various strategies involving different theorems, such as L'Hôpital's rule which deals with indeterminate forms, or theorems relating to the continuity of a function. Each theorem has its own conditions and applications, but what's common among them is their foundational logic: they provide structured pathways to navigate the complex landscape of functions and their limits. Like detectives following different lines of enquiry, mathematicians use these theorems to home in on the truth about a function's behavior, closing in on the elusive number that is the limit—a critical pursuit in the world of calculus.