It looks like you're asking for the limit of an expression as \( t \to 0^+ \). The expression is:

\[
\lim_{t \to 0^+} \frac{x \ln(9x) - x}{3t}
\]

I assume \( x \) and \( t \) are independent variables. First, let's break this problem down step by step.

Since you are taking the limit as \( t \to 0^+ \), and the expression is in terms of \( t \) only in the denominator, we need to focus on the behavior of \( x \) in terms of \( t \).

### Step 1: Expression analysis
The given expression is \( \frac{x \ln(9x) - x}{3t} \). For the limit as \( t \to 0^+ \), the behavior of \( x \) in terms of \( t \) needs to be understood to see whether this limit is well-defined.

However, without knowing the relationship between \( x \) and \( t \), the limit cannot be simplified directly. Typically, in limits of this type, one might make an assumption about how \( x \) behaves as \( t \to 0^+ \).

1. **Case 1: \( x \) is a constant** — If \( x \) does not depend on \( t \), the limit would be undefined, since the denominator goes to zero while the numerator stays constant, leading to an infinite value.
   
2. **Case 2: \( x = f(t) \), where \( x \) is a function of \( t \)** — If there is a specific relationship between \( x \) and \( t \), such as \( x = t \) or a similar functional form, the limit might yield a more meaningful result.

Could you clarify if \( x \) is a constant or a function of \( t \)? That would help determine how to proceed with the limit.

---
Would you like details on how to compute limits involving logarithmic functions? Here are some related questions you might find useful:

1. What is the limit of \( \lim_{x \to 0^+} x \ln(x) \)?
2. How does L'Hopital's Rule apply to limits of indeterminate forms?
3. What are some common techniques for solving limits involving logarithms?
4. Can you give an example where a function tends to infinity as its input tends to zero?
5. What is the significance of asymptotic behavior in calculus?

**Tip**: When handling logarithmic limits, always check if the form is indeterminate (e.g., \( 0 \times \infty \)) to decide if techniques like L'Hopital’s Rule or substitution are required.

Explore the limit of the expression lim t → 0+ (xln(9x) − x) / 3t using L'Hopital's Rule and logarithmic properties. This problem demonstrates key calculus techniques for handling indeterminate forms. Suitable for students in undergraduate calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation