Problem 38 - APEX Calculus

Question

Use the Squeeze Theorem, where appropriate, to evaluate the given limit. $$ \lim _{x \rightarrow 3} f(x), \text { where } 6 x-9 \leq f(x) \leq x^{2} $$

Step-by-Step Solution

Answer

The limit is 9.

Step 1: Set Up the Known Inequality

We start with the given inequality \( 6x - 9 \leq f(x) \leq x^2 \). The Squeeze Theorem can be applied if we can evaluate the limits of the bounding functions \( 6x - 9 \) and \( x^2 \) as \( x \to 3 \) and they are equal.

Step 2: Calculate Limit of the Lower Bound

Evaluate \( \lim_{x \to 3} (6x - 9) \).Substitute \( x = 3 \) into the expression:\[ 6(3) - 9 = 18 - 9 = 9 \]

Step 3: Calculate Limit of the Upper Bound

Evaluate \( \lim_{x \to 3} x^2 \).Substitute \( x = 3 \) into the expression:\[ (3)^2 = 9 \]

Step 4: Apply the Squeeze Theorem

According to the Squeeze Theorem, if \( \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \) and \( g(x) \leq f(x) \leq h(x) \) for all \( x \) near \( a \), then \( \lim_{x \to a} f(x) = L \).In this case, since both limits evaluated in Steps 2 and 3 are 9, we use them to conclude:\[ \lim_{x \to 3} f(x) = 9 \]

Key Concepts

Limits

When working with limits, you are essentially trying to understand the behavior of a function as it approaches a certain point. The notation \( \lim_{x \to a} f(x) \) expresses the limit of \( f(x) \) as \( x \) approaches \( a \).
This is fundamental in calculus because it lays the groundwork for other concepts such as derivatives and integrals. In our exercise, you can see this in action as we evaluate \( \lim_{x \to 3} f(x) \).
The key to solving limits, especially when using the Squeeze Theorem, is to establish what happens as \( x \) gets infinitely close to 3. When you substitute 3 into the bounding functions (as shown in the solution), you quickly pinpoint the target value that \( f(x) \) should approach, thanks to the equal limits of the bounding functions at that point.
If limits sometimes seem abstract, think of them as asking: what does \( f(x) \) want to be when \( x \) comes knocking at the door of 3?

Bounding Functions

Bounding functions refer to the upper and lower functions that encompass another function like guards. These functions are vital in applying the Squeeze Theorem.
In our example problem, we have two bounding functions: \( 6x - 9 \) and \( x^2 \). These functions serve as a safety net to ensure that \( f(x) \) does not go astray as \( x \) approaches 3.
By evaluating the limits of these bounding functions separately, you can determine the limit of the function squeezed in between. Both need to hit the same target value to effectively 'squeeze' the third function's limits to that same value.

• Lower bound: \(6x - 9\) hits the value 9 at \( x = 3 \).
• Upper bound: \(x^2\) also reaches the value 9 at \( x = 3 \).

This dual-sided bounding assures the Squeeze Theorem applies.

Calculus

Calculus is more than just a study of limits. It’s a powerful mathematical tool that allows us to model and solve complicated problems that involve change. The Squeeze Theorem is a clever calculus technique,
which states if a function is sandwiched between two others that share the same limit at a particular point, then it must share that limit too.
Calculus tools like these are woven into many aspects of science and engineering as they let us calculate values that might otherwise seem tricky to pin down.

• The derivative, which is about finding slopes and rates of change, starts with understanding limits.
• The integral, which measures areas under curves, also uses limits.
• Real-world applications span physics to economics, showing the importance of grasping these foundational concepts.

Using the Squeeze Theorem efficiently reduces the need for complex calculations by simplifying the process into evaluating easier limits for the bounding functions.