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.

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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.