Question

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 4} \sqrt{x} $$

Short answer

The limit as \(x\) approaches \(4\) for \(\sqrt{x}\) is \(2\). This limit has been verified by using the epsilon-delta definition of the limit.

Step-by-step solution

Finding the Limit

Firstly, we need to find the limit, \(L\), of the function as \(x\) approaches \(4\). This is done by substituting \(4\) into the function \(\sqrt{x}\), therefore \(L = \sqrt{4} = 2\).

Setting up the Epsilon-Delta Definition

Next, we need to show that for every \(\epsilon\), we can find a \(\delta\) such that when \(0 < |x-4| < \delta\), it follows that \(|\sqrt{x} - 2| < \epsilon\). We square both sides of the inequality to remove the square root, resulting in \(|x - 4| < (\epsilon / 2)^2\). So, we can choose \(\delta = (\epsilon/2)^2\).

Verifying the Epsilon-Delta Definition

To verify the definition, we substitute our chosen \(\delta\) back into the original inequality. If \(0 < |x-4| < (\epsilon/2)^2\), then \(|\sqrt{x} - 2| < \epsilon\). Thus, we have verified that for this function and limit, for any \(\epsilon > 0\), there exists a \(\delta = (\epsilon/2)^2\) such that the definition holds, proving that the limit is \(2\).

Key concepts

Square Root Function

The square root function is one of the fundamental mathematical functions, often represented as \( f(x) = \sqrt{x} \). Its primary feature is that it returns a value such that when squared, it equals the input number \(x\). For example, \(\sqrt{4}\) equals \(2\) because \(2^2 = 4\). This function reveals interesting properties, particularly in calculus, due to its non-linear nature. It is distinct from polynomial functions, characterized by growth that slows as \(x\) increases.

• The square root function is defined only for non-negative values of \(x\), indicating the set of real numbers, \([0, \infty)\).
• It is an increasing function, meaning as \(x\) increases, \(\sqrt{x}\) also increases.
• At \(x = 0\), \(\sqrt{x}\) reaches its minimum value, 0.

Understanding the behavior of the square root function is crucial as it forms part of many more complex functions and calculations in the real world.

Limit Proof

In calculus, the limit is a way to describe the behavior of a function as its input approaches a certain value. The proof of a limit using the epsilon-delta definition is the formal method to do this. For the function \( \lim_{x \to 4} \sqrt{x} \), our task is to demonstrate that as \(x\) gets closer to \(4\), \(\sqrt{x}\) gets closer to \(2\).

• First, find the limit \(L\) by direct substitution: \( L = \sqrt{4} = 2 \).
• To prove that \(L\) is actually the limit, we use the epsilon-delta definition.

You define \( \epsilon \) as any positive number that represents how close you want \(\sqrt{x}\) to be to \(2\). Next, you find a corresponding \(\delta\), such that whenever \(0 < |x - 4| < \delta\), it leads to \(|\sqrt{x} - 2| < \epsilon\). This involves squaring both sides as shown in the solution, resulting in our choice for \(\delta\): \((\epsilon/2)^2\).

Calculus Concepts

Calculus is a branch of mathematics that helps us understand changes and motion. It includes concepts and methods for finding limits, derivatives, and integrals. The epsilon-delta definition of a limit is one such fundamental concept, providing rigor to intuitive ideas about approaching values.

• Limits explore what happens as the input of a function approaches a particular point, revealing insights about function behavior.
• This is crucial for understanding continuity and differentiability, features of calculus that describe how smooth or jagged a function graph is.
• The epsilon-delta proof ensures that we do not rely on guesswork, but rather verifies through precise logic how limits are truly defined.

Engaging with these core calculus concepts deeply enhances the comprehension of how functions behave in the small-scale, enabling learners to tackle more complex mathematical problems competently.

Approaching a Value

Approaching a value in calculus means observing what happens to a function as the input nears a specific number. It embodies the core idea of limits, where we find out what value a function's output is getting closer to as the input approaches some value. This concept is crucial in the formal definition of limits.

• The rule \( \lim_{x \to 4} \sqrt{x} = 2 \) means as \(x\) gets very close to \(4\), \(\sqrt{x}\) gets very close to \(2\).
• The function doesn't need to reach the value; it only matters the output approaches it as input approaches the target value.
• This idea is applied in all kinds of functions and paves the way for understanding derivatives, which describe how functions change momentarily.

Grasping how functions approach and dependably draw near a specific value aids in making the leap from algebra to advanced calculus, illuminating why and how calculus works.