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.