Problem 35 - Calculus Early Transcendentals

Question

If \(4 x-9 \leqslant f(x) \leqslant x^{2}-4 x+7\) for \(x \geqslant 0,\) find \(\lim _{x \rightarrow 4} f(x)\)

Step-by-Step Solution

Answer

The limit is 7.

Step 1: Determine the boundary functions at the limit point

To find \(\lim _{x \rightarrow 4} f(x)\), we need to examine the behavior of the boundary functions \(4x - 9\) and \(x^2 - 4x + 7\) at \(x = 4\). First, calculate \(4x - 9\) at \(x = 4\): \(4(4) - 9 = 16 - 9 = 7\). Next, calculate \(x^2 - 4x + 7\) at \(x = 4\): \(4^2 - 4\times4 + 7 = 16 - 16 + 7 = 7\).

Step 2: Apply the Squeeze Theorem

The Squeeze Theorem states that if \(g(x) \leq f(x) \leq h(x)\) for all \(x\) in some interval around \(a\) (except possibly at \(a\) itself), and if \(\lim _{x \to a} g(x) = \lim _{x \to a} h(x) = L\), then \(\lim _{x \to a} f(x) = L\). Here, \(4x - 9\) and \(x^2 - 4x + 7\) both approach 7 as \(x\) approaches 4. Thus, \(f(x)\) is squeezed to the limit of 7 as well.

Step 3: Conclude the limit of the function

Since both boundary functions have the same limit at \(x = 4\), the limit of \(f(x)\) as \(x\) approaches 4 is also 7. Thus, \(\lim _{x \rightarrow 4} f(x) = 7\).

Key Concepts

Limits

The concept of limits is fundamental in calculus and helps us understand the behavior of functions as they approach a certain point. When we talk about the limit of a function, we're essentially asking what value the function is getting closer to as we approach a certain point on the x-axis. Here, we're interested in the limit of the function \(f(x)\) as \(x\) approaches 4.
With limits, we're often interested in the behavior of a function around a point, rather than exactly at it. This is particularly useful when we have functions that may not be defined at a certain point or when dealing with continuous functions like in this example. Calculating limits often involves substituting the point into the function, given that the function is continuous and well-behaved around that value.
In the context of the exercise, we use the Squeeze Theorem to determine the limit. This theorem is particularly useful when a function is "trapped" between two other functions whose limits we can easily calculate.

Boundary Functions

Boundary functions help us in using the Squeeze Theorem by providing two functions that "squeeze" the original function from above and below. In this exercise, the boundary functions for \(f(x)\) are \(g(x) = 4x - 9\) and \(h(x) = x^2 - 4x + 7\).
Evaluating these boundary functions at \(x = 4\), we find that both reach the value of 7. This is crucial because it means that \(f(x)\) must also be approaching the same value as \(x\) approaches 4. The key property here is that the inequality holds around a neighborhood of 4, allowing us to apply the Squeeze Theorem effectively.
The boundary functions offer us a straightforward way to handle the often complex behavior of functions, simplifying the process of finding limits for a function within a given interval.

Function Behavior at a Point

Understanding a function's behavior at a given point involves analyzing how the function behaves as we get closer and closer to the point of interest. This behavior often tells us a lot about the function's continuity, potential limits, and possible discontinuities.
In this problem, as \(x\) approaches 4, we're keen on how \(f(x)\) behaves. Thanks to the boundary functions, we know that \(f(x)\) is constrained between two specific values, making it easier to predict its behavior. Both boundary values become the same at \(x = 4\), which simplifies the problem dramatically.
Thus, the function \(f(x)\) also behaves similarly to the boundary functions when \(x\) is near 4, driving home the utility of using the Squeeze Theorem to understand function behavior in settings where direct evaluation might be challenging due to complexities in the function's form.