Question

Using the Squeeze Theorem In Exercises 89 and 90 , use the Squeeze Theorem to find \(\lim _{x \rightarrow c} f(x)\) $$ \begin{array}{l}{c=0} \\ {4-x^{2} \leq f(x) \leq 4+x^{2}}\end{array} $$

Short answer

\(\lim_{x \rightarrow 0} f(x) = 4\)

Step-by-step solution

Find the limits of the bounding functions

Calculate the limit of the functions \(4-x^{2}\) and \(4+x^{2}\) as x approaches 0. The result will be the same for both functions since the term \(x^{2}\) becomes 0 as x approaches 0. E.g. \(\lim_{x \rightarrow 0} (4-x^{2}) = 4-0^{2} = 4\). Similarly, \(\lim_{x \rightarrow 0} (4+x^{2}) = 4+0^{2}=4\)

Apply the Squeeze Theorem

The Squeeze Theorem states that if function g(x) is bounded by functions f(x) and h(x) and the limits of f(x) and h(x) as x approaches c are equal, then the limit of g(x) as x approaches c is also equal to this common limit. In this case, the limit of both \(4-x^{2}\) and \(4+x^{2}\) as x approaches 0 is 4. Therefore, by the Squeeze Theorem, the limit of f(x) as x approaches 0 is also 4.

Final Answer

So, from the Squeeze theorem, we can conclude that \(\lim_{x \rightarrow 0} f(x) = 4\)

Key concepts

Understanding the Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a particular value. Essentially, it shows us what value the function is tending towards without necessarily reaching it. This concept is essential in calculus as it helps understand continuity and the behavior of functions near certain points.

For example, in the exercise, we found the limit of the bounding functions, which were in the form of simple polynomials. Both statements showed that as the variable \(x\) approaches 0, the functions \(4-x^2\) and \(4+x^2\) tend towards 4. By calculating these limits, we decipher the behavior of the function \(f(x)\) which is squeezed between them.

Exploring Bounding Functions

In the Squeeze Theorem, bounding functions play a crucial role. These are two functions that create an upper and lower "bound" for another function. They essentially "squeeze" the targeted function between them, guiding its behavior as \(x\) approaches a specific value.

In our exercise, we used the bounding functions \(4-x^2\) and \(4+x^2\). These functions are especially useful because their limits were easy to calculate. By setting \(4-x^2 \leq f(x) \leq 4+x^2\), we know that whatever happens to the bounding functions will also happen to \(f(x)\) as \(x\) approaches 0. This creates a clear path to finding the limit of \(f(x)\) using more manageable components.

Application of Calculus Through the Squeeze Theorem

Calculus often tackles changing quantities, and understanding how these changes behave is crucial. The Squeeze Theorem is one application within calculus that allows us to determine a function's limit, even if directly evaluating the function is complex or impossible.

In applying the theorem, we begin with bounding our function between two simpler ones. As shown in the calculation steps, by ensuring both bounding functions lead to a limit of 4, we conclude the same for \(f(x)\). This method shows a fundamental calculus principle: simplifying complex problems using known techniques.The Squeeze Theorem is a powerful tool because it connects simpler, calculated limits into understanding more complicated scenarios.