Problem 20 - Calculus Early Transcendentals

Question

Prove the statement using the \(\varepsilon, \varepsilon\) definition of limit. \(\lim _{x \rightarrow 6}\left(\frac{x}{4}+3\right)=\frac{9}{2}\)

Step-by-Step Solution

Answer

Choose \(\delta = 4\varepsilon\) for any \(\varepsilon > 0\) to prove the limit.

Step 1: Understand the Epsilon-Delta Definition of a Limit

To prove \(\lim _{x \rightarrow 6}\left(\frac{x}{4}+3\right)=\frac{9}{2}\), we need to prove that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - 6| < \delta\), then \(\left|\left(\frac{x}{4}+3\right)-\frac{9}{2}\right| < \varepsilon\). This involves showing that the function \(f(x) = \frac{x}{4} + 3\) gets arbitrarily close to \(\frac{9}{2}\) as \(x\) approaches \(6\).

Step 2: Simplify the Limit Expression

Compute \(f(x) - \frac{9}{2}\), which should be less than \(\varepsilon\):\[\left|\left(\frac{x}{4}+3\right)-\frac{9}{2}\right| = \left|\frac{x}{4} + 3 - \frac{9}{2}\right|\]Since \(\frac{9}{2} = \frac{18}{4}\), the expression becomes:\[\left|\frac{x}{4} + 3 - \frac{18}{4}\right| = \left|\frac{x}{4} - \frac{6}{4}\right| = \left|\frac{x-6}{4}\right|\]This simplifies down to \(\frac{|x - 6|}{4}\).

Step 3: Choose Delta in Terms of Epsilon

We need \(\frac{|x - 6|}{4} < \varepsilon\). To solve for \(\delta\), multiply both sides by 4:\[|x - 6| < 4\varepsilon\]So, we choose \(\delta = 4\varepsilon\). This means if \(0 < |x - 6| < 4\varepsilon\), then \(\left|\frac{x}{4} + 3 - \frac{9}{2}\right| < \varepsilon\).

Step 4: Conclusion

Since we've shown that for every \(\varepsilon > 0\), choosing \(\delta = 4\varepsilon\) ensures \(|x - 6| < \delta\) implies \(\left|\left(\frac{x}{4}+3\right)-\frac{9}{2}\right| < \varepsilon\), the limit is proven. Therefore, \(\lim _{x \rightarrow 6}\left(\frac{x}{4}+3\right)=\frac{9}{2}\) holds true by the \(\varepsilon, \delta\) definition of a limit.

Key Concepts

epsilon-delta proof

An epsilon-delta proof is a formal method used to establish the limit of a function rigorously. The proof is structured to confirm that a function approaches a specific value as the input approaches a specific point. Let's break down the components:

• Epsilon (\(\varepsilon\)): This is a positive number, no matter how small, representing how close we want the function's value to be to the limit.
• Delta (\(\delta\)): This is also a positive number, indicating that the input values must be within this distance from the target point, except for exactly the point itself.

For our function \(f(x) = \frac{x}{4} + 3\) as \(x\) approaches \(6\), the proof requires us to find a \(\delta\) for every \(\varepsilon\) such that when \(0 < |x - 6| < \delta\), the inequality \(\left|\frac{x}{4} + 3 - \frac{9}{2}\right| < \varepsilon\) holds true. This establishes that the function's value is close to \(\frac{9}{2}\) when \(x\) is near \(6\).
This systematic approach ensures there are no exceptions beyond the desired tolerance level, thus confirming the limit precisely within the epsilon-delta framework.

calculus limits

Calculus limits are foundational in understanding how functions behave as they approach certain points. Limits describe the value that a function approaches as the input approaches some value. They are essential for defining continuity, derivatives, and integrals.
To comprehend limits better, consider a simple function \(f(x) = \frac{x}{4} + 3\). As \(x\) gets closer and closer to \(6\), the function's value approaches \(\frac{9}{2}\). This behavior is what we represent formally using limits.

• The "approaching" aspect: In the expression \(\lim_{x \rightarrow c} f(x) = L\), \(x\) isn't required to 'reach' \(c\) for the limit to exist; rather, it must get arbitrarily close.
• Diverse scenarios: Limits apply whether \(f(x)\) rises or drops as \(x\) nears \(c\), or even if it's undefined precisely at \(c\)

Calculus limits allow us to tackle functions at points of discontinuity and explore their local behavior. They make it possible to capture the instantaneous rate of change and the concept of areas under curves—central to calculus principles.

limit of a function

The limit of a function expresses where a function seems to be heading as the input nears a particular point. It provides a conceptual understanding of the function's behavior in the neighborhood of that point.
For example, observe \(f(x) = \frac{x}{4} + 3\) as \(x\) approaches \(6\). Evaluating the expression shows:

• As \(x\) closely approximates \(6\) from either side, the function output nudges closer to \(\frac{9}{2}\)
• This 'nudging' is quantified mathematically using epsilon-delta definitions to prove this behavior rigorously.
• The limit helps pinpoint this converging value despite any jumps or undefined points at \(x = 6\)

Limits are critical for establishing a foundational understanding of mathematical analysis, especially for defining slope (derivatives) at a given point and calculating exact areas (integrals) for complex functions. Recognizing the limit itself links to predicting or explaining function behaviors at essential points in their domain.