Let's solve the limit $\lim_{x \to 2} f(x) = 5$ in great detail.

Step 1: Understanding the Limit Concept

The statement $\lim_{x \to 2} f(x) = 5$ means that as $x$ approaches 2, the value of $f(x)$ gets arbitrarily close to 5, regardless of whether $f(x)$ is actually equal to 5 when $x = 2$.

Step 2: Formal Definition of a Limit

To rigorously prove $\lim_{x \to 2} f(x) = 5$, we use the $\epsilon$-$\delta$ definition of a limit. According to this definition, for every $\epsilon > 0$ (no matter how small), there exists a $\delta > 0$ such that whenever $0 < |x - 2| < \delta$, it follows that $|f(x) - 5| < \epsilon$.

Here’s what each part means:

• $\epsilon$ represents how close $f(x)$ is to 5.
• $\delta$ represents how close $x$ is to 2.

Step 3: Analyzing the Function $f(x)$

To proceed, we usually need to know the specific form of $f(x)$. However, if $f(x)$ is not given explicitly, we'll discuss the general approach.

Case 1: $f(x)$ is Continuous at $x = 2$

If $f(x)$ is continuous at $x = 2$, then: $\lim_{x \to 2} f(x) = f(2)$ Given $\lim_{x \to 2} f(x) = 5$, it follows that $f(2) = 5$.

In this case, the solution is straightforward because continuity implies that as $x$ approaches 2, $f(x)$ will approach $f(2)$, which is given as 5.

Case 2: $f(x)$ is Discontinuous at $x = 2$

If $f(x)$ is not continuous at $x = 2$, the value of $f(2)$ could be different from 5, or $f(2)$ might not even be defined. In this case, we must rely solely on the behavior of $f(x)$ as $x$ approaches 2 from both sides.

Step 4: Verifying with $\epsilon$-$\delta$ Definition

Let's apply the $\epsilon$-$\delta$ definition to verify the limit.

1. Given: $\epsilon > 0$, we need to find a corresponding $\delta > 0$ such that: $0 < |x - 2| < \delta \implies |f(x) - 5| < \epsilon$

2. Choose $\delta = \delta(\epsilon)$ such that this condition holds. The exact choice of $\delta$ depends on the specific function $f(x)$, but the goal is to ensure that as $x$ gets within $\delta$ units of 2, the value of $f(x)$ is within $\epsilon$ units of 5.

Step 5: Conclusion

If you can find such a $\delta$ for any $\epsilon > 0$, then by the $\epsilon$-$\delta$ definition, the limit $\lim_{x \to 2} f(x) = 5$ is confirmed.

Step 6: Example (Hypothetical Function)

If $f(x) = 2x + 1$, then: $\lim_{x \to 2} f(x) = \lim_{x \to 2} (2x + 1) = 2(2) + 1 = 5$ This matches our given limit, and in this case, $\delta = \epsilon / 2$ would work.

Recap and Reflection

The limit $\lim_{x \to 2} f(x) = 5$ means that as $x$ approaches 2, the function $f(x)$ approaches 5, regardless of $f(2)$. The $\epsilon$-$\delta$ definition is a precise mathematical way to prove this behavior.

Would you like more details on a specific part, or do you have any questions?

Related Questions:

1. What happens if the limit at a point does not exist?
2. How does the concept of one-sided limits relate to the overall limit?
3. Can the limit of a function exist if the function is not defined at that point?
4. How can we determine limits of more complex functions, such as rational functions?
5. What is the relationship between the limit and the derivative of a function?

Tip: When solving limits, always consider the continuity of the function and the possibility of one-sided limits!