Enter the exact answer. If the limit does not exist, enter DNE.
 
limx→3+x2x2−9

To solve the given limit:

\[
\lim_{x \to 3} \frac{x}{x^2 - 9}
\]

we start by analyzing and simplifying the expression.

### Step 1: Factor the denominator

The denominator \(x^2 - 9\) is a difference of squares and can be factored as:

\[
x^2 - 9 = (x - 3)(x + 3)
\]

### Step 2: Rewrite the limit expression

Substitute the factorized form of the denominator into the limit expression:

\[
\lim_{x \to 3} \frac{x}{(x - 3)(x + 3)}
\]

### Step 3: Check for an indeterminate form

Substituting \(x = 3\) into the limit expression gives:

\[
\frac{3}{(3 - 3)(3 + 3)} = \frac{3}{0 \cdot 6} = \frac{3}{0}
\]

Since we have a non-zero numerator and a zero denominator, this indicates the limit may approach infinity or negative infinity. However, this isn't an indeterminate form like \(\frac{0}{0}\).

### Step 4: Analyze the behavior around \( x = 3 \)

To better understand the behavior of the limit, let's consider values of \(x\) approaching 3 from both sides:

- **As \(x\) approaches 3 from the right (\(x > 3\)):**
    \[
    \lim_{x \to 3^+} \frac{x}{(x - 3)(x + 3)} = \lim_{x \to 3^+} \frac{x}{(x - 3)(x + 3)}
    \]
    Here, \((x - 3)\) is positive, and \((x + 3)\) is positive, so the whole expression approaches \( +\infty \).

- **As \(x\) approaches 3 from the left (\(x < 3\)):**
    \[
    \lim_{x \to 3^-} \frac{x}{(x - 3)(x + 3)} = \lim_{x \to 3^-} \frac{x}{(x - 3)(x + 3)}
    \]
    Here, \((x - 3)\) is negative, and \((x + 3)\) is positive, so the whole expression approaches \( -\infty \).

Since the limits from the left and the right do not match (one approaches \(+\infty\) and the other approaches \(-\infty\)), the two-sided limit does not exist.

### Final Answer

\[
\lim_{x \to 3} \frac{x}{x^2 - 9} = \text{DNE}
\]

Would you like more details on any step or have any questions?

Here are some related questions:

1. What does it mean for a limit to not exist?
2. How can we analyze limits approaching infinity?
3. What are one-sided limits, and how are they different from two-sided limits?
4. How can limits be used to define the continuity of a function?
5. What are the differences between removable and non-removable discontinuities?

**Tip:** Always check the behavior of the function from both sides when dealing with limits approaching a point of discontinuity.

Learn how to evaluate the limit lim(x -> 3) [x / (x^2 - 9)] step-by-step with a detailed solution. Understand the concept of limits and indeterminate forms. Suitable for college-level students and anyone interested in calculus and mathematical analysis.

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