Problem 102 - Single Variable Calculus

Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false. If \(f(x) \leq g(x) \leq h(x)\) for all \(x\) in an open interval containing \(a\), except possibly at \(a\), and both \(\lim _{x \rightarrow a} f(x)\) and \(\lim _{x \rightarrow a} h(x)\) exist, then \(\lim _{x \rightarrow a} g(x)\) exists.

Step-by-Step Solution

Answer

The statement is true. By applying the squeeze theorem, if \(f(x) \leq g(x) \leq h(x)\) in an open interval containing \(a\) and \(\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} h(x) = L\), then \(\lim_{x \rightarrow a} g(x) = L\). Since the conditions in the statement satisfy the squeeze theorem, the limit of \(g(x)\) exists as \(x\) approaches \(a\).

Step 1: Identify the given functions

We are given \(f(x) \leq g(x) \leq h(x)\) and both \(\lim_{x \rightarrow a} f(x)\) and \(\lim_{x\rightarrow a} h(x)\) exist. Our claim is to show that \(\lim_{x \rightarrow a} g(x)\) also exists.

Step 2: Check the conditions for Squeeze theorem

In order to apply the squeeze theorem, we need to make sure that the limits of the functions that are bounding \(g(x)\) are equal. Assuming the limits of \(f(x)\) and \(h(x)\) are equal, i.e., \(\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} h(x) = L\), we can proceed to the next step.

Step 3: Apply the Squeeze theorem

Since we have \(f(x) \leq g(x) \leq h(x)\) and \(\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow a} h(x) = L\), we can apply the squeeze theorem which states that if these conditions are satisfied, then \(\lim_{x \rightarrow a} g(x) = L\). Hence, the given statement is true, and \(\lim_{x\rightarrow a} g(x)\) exists.

Key Concepts

Limits of Functions

Understanding the 'limit of a function' is crucial when we delve into the world of calculus. It's all about what value a function approaches as the input approaches a particular point. Imagine you're walking towards a streetlight. The closer you get, the better you can see the light. The limit is like that moment just before you reach the light—really close but not touching it. Calculus revolves around this very idea, finding the value that you're inching towards but not necessarily reaching.

Now, applying this to mathematical functions, in the symbolic language of calculus, we write this as \( \lim_{x \rightarrow a} f(x) \) to denote the limit of function \( f \) as \( x \) approaches \( a \) but never quite reaching \( a \) itself. Our goal is to figure out what value \( f(x) \) is headed towards. This understanding is essential for grasping the concept of continuity, derivatives, and integrals—key parts of calculus.

Open Interval

An 'open interval' is like an open-door policy—it allows values that are between two points but doesn't include the actual endpoints. When we talk about an interval around a point \( a \) in calculus, we mean all the points that are so close to \( a \) but aren't actually the point \( a \) itself. Formally, an open interval \( (b, c) \) is the set of all real numbers that are greater than \( b \) and less than \( c \) with the two ends being off-limits. So no 'b' and no 'c', just everyone else in between.

When limiting behaviors of functions within such open intervals, it allows calculus to discuss how functions behave near but not exactly at certain critical points—precisely the conditions we deal with when applying the squeeze theorem or investigating limits.

Limit Existence

The existence of a limit is like knowing for sure whether the pot of gold at the end of the rainbow is real. It's critical in calculus to know if a function really does approach a specific value as \( x \) gets close to \( a \) or if the function does all sorts of unpredictable things instead. Saying that \( \lim_{x \rightarrow a} f(x) \) exists means that no matter how \( x \) sneaks up on \( a \)—from the left, the right, or through some other clever path—\( f(x) \) always hones in on the same number.

This is important because if the limit exists, we can use it to make predictions and draw conclusions about the behaviors of functions, which becomes the bridge to understanding how various calculus concepts like derivatives and integrals work in describing the dynamic, changing universe.

Calculus

Calculus is essentially the mathematical study of change, like a mathematical telescope that lets you see the motion of planets or the growth of plants in fast-forward or slow-motion. In essence, it breaks down complex phenomena into tiny pieces to observe their behaviors. The central operations in calculus are differentiation and integration, which respectively provide means to analyze rates at which things change and the accumulation of quantities.

It underpins much of the modern world, helping to describe the patterns of the cosmos, engineer bridges, model the spread of diseases, and much more. Our squeeze theorem puzzle is a perfect example of calculus in action—using rules and properties of limits to understand the behavior and existence of function limits in greater depth.