Lim of x/√x²-9 as x approaches to -3^-

To solve the limit of \( \frac{x}{\sqrt{x^2 - 9}} \) as \( x \to -3^{-} \), we proceed step-by-step:

### Step 1: Examine the behavior of the function as \( x \to -3 \) from the left.

We have the expression:
\[
f(x) = \frac{x}{\sqrt{x^2 - 9}}.
\]

### Step 2: Analyze the denominator.

In the denominator, \( \sqrt{x^2 - 9} \), the expression \( x^2 - 9 \) becomes 0 when \( x = \pm 3 \). So, near \( x = -3 \), we have:
\[
x^2 - 9 = (-3)^2 - 9 = 9 - 9 = 0.
\]
As \( x \to -3^{-} \), the expression \( x^2 - 9 \) is positive but very small, so \( \sqrt{x^2 - 9} \) approaches 0 from the positive side.

### Step 3: Analyze the numerator.

The numerator is simply \( x \), and as \( x \to -3^{-} \), \( x \) approaches -3.

### Step 4: Evaluate the limit.

Now, we need to consider the behavior of the whole expression:
\[
\frac{x}{\sqrt{x^2 - 9}}.
\]
As \( x \to -3^{-} \):
- The numerator approaches \( -3 \).
- The denominator approaches 0 from the positive side (since \( x^2 - 9 \) is positive and small near \( x = -3 \)).

Thus, the fraction \( \frac{-3}{\text{small positive number}} \) becomes very large and negative. Therefore, the limit is:
\[
\lim_{x \to -3^{-}} \frac{x}{\sqrt{x^2 - 9}} = -\infty.
\]

### Final Answer:
\[
\lim_{x \to -3^{-}} \frac{x}{\sqrt{x^2 - 9}} = -\infty.
\]

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:
1. What is the limit of \( \frac{x}{\sqrt{x^2 - 9}} \) as \( x \to 3^+ \)?
2. How do you evaluate limits involving radicals and absolute values?
3. Can we simplify expressions with radicals to find limits more easily?
4. What happens to the function \( \frac{x}{\sqrt{x^2 - 9}} \) at \( x = 0 \)?
5. What are one-sided limits, and why are they important?

**Tip:** When evaluating limits with square roots, always check the sign of the expression under the square root to ensure you are considering valid real numbers.

Learn how to solve the limit of the expression x/√(x² - 9) as x approaches -3 from the left. This problem involves analyzing radicals and one-sided limits, with a detailed solution showing the behavior of the function near the critical point. Ideal for students in Grades 11-12.

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