To find the limit of the expression \(\lim_{x \to 3^-} \frac{x}{3 - x}\), we need to evaluate how the function behaves as \(x\) approaches 3 from the left.

1. **Substitute values slightly less than 3**:
   - As \(x\) approaches 3 from the left (i.e., values slightly less than 3, like 2.9, 2.99, etc.), the denominator \(3 - x\) becomes a small positive number approaching 0.
   - The numerator \(x\) approaches 3 as well.

2. **Behavior of the function**:
   - As \(x\) approaches 3 from the left, \(3 - x\) approaches 0 from the **positive** side. Thus, the fraction \(\frac{x}{3 - x}\) grows without bound, heading towards **positive infinity**.

Thus, the limit is:
\[
\lim_{x \to 3^-} \frac{x}{3 - x} = +\infty
\]

Would you like more details or clarification on any part of this?

Here are 5 related questions:
1. What happens to the limit when \( x \to 3^+ \)?
2. How do limits behave when approaching from the right versus from the left?
3. Can limits be used to define discontinuities?
4. What are some examples of limits resulting in negative infinity?
5. How do asymptotes relate to limits?

**Tip**: When calculating limits, always check if the function behaves differently from the left-hand and right-hand sides.

lim x approaches 3 from the left of x / (3 - x)

Discover how to evaluate the limit of x/(3 - x) as x approaches 3 from the left. This problem covers essential concepts such as one-sided limits, asymptotic behavior, and the behavior of functions near discontinuities. Ideal for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation