猜猜她是谁

158516***

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朱***

15851667124 发表于 2025-9-22 19:04

Alright, let's go step by step.

---

## **1. Understanding the problem**

We are given a function  
\[
f(x) = \frac{x^2 - 4}{x - 2}
\]
and we want to find the limit:
\[
\lim_{x \to 2} f(x)
\]
and then determine which of the given statements is true.

The statements are:

1. \(\lim_{x \to 2} f(x)\) does not exist because \(f(2)\) is undefined.
2. \(\lim_{x \to 2} f(x) = 4\) because \(\lim_{x \to 2} f(x)\) depends on values of \(x\) close to 2 but not equal to 2.
3. \(\lim_{x \to 2} f(x)\) does not exist because division by zero is undefined.

---

## **2. Simplify \(f(x)\)**

\[
f(x) = \frac{x^2 - 4}{x - 2}
\]
Factor numerator:
\[
x^2 - 4 = (x - 2)(x + 2)
\]
So for \(x \neq 2\):
\[
f(x) = \frac{(x - 2)(x + 2)}{x - 2} = x + 2
\]

Thus:
\[
f(x) = x + 2 \quad \text{for all } x \neq 2
\]

---

## **3. Compute the limit**

\[
\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 2 + 2 = 4
\]

So the limit exists and equals 4, even though \(f(2)\) is undefined.

---

## **4. Evaluate each statement**

**Statement 1:**  
"\(\lim_{x \to 2} f(x)\) does not exist because \(f(2)\) is undefined."  
This is false — the limit depends on values near \(x = 2\), not at \(x = 2\). A function can have a limit at a point where it is undefined.

**Statement 2:**  
"\(\lim_{x \to 2} f(x) = 4\) because \(\lim_{x \to 2} f(x)\) depends on values of \(x\) close to 2 but not equal to 2."  
This is true — we simplified \(f(x)\) to \(x+2\) for \(x \neq 2\), so indeed near \(x=2\) the function behaves like \(x+2\), giving limit 4.

**Statement 3:**  
"\(\lim_{x \to 2} f(x)\) does not exist because division by zero is undefined."  
This is false — division by zero at \(x=2\) means \(f(2)\) is undefined, but the limit can still exist.

---

## **5. Conclusion**

The correct statement is **Statement 2**.

\[
\boxed{2}
\]   

--来源：越嘉艺术网