Question

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 2^{-}} \ln \left[x^{2}(3-x)\right] $$

Short answer

The limit of the function as \(x\) approaches 2 from the left is \(\ln [2]\).

Step-by-step solution

Understanding the Limit Notation

The notation \(x \rightarrow 2^{-}\) means that \(x\) is approaching 2 from its left side, or from the negative side. This is known as one-sided limit.

Substituting the limit to the given function

Substitute \(x = 2\) into the function \(x^{2}(3-x)\). Our function simplifies to: \(2^{2}(3-2) = 2\). Then, take the natural logarithm (since our function is \(\ln [x^{2}(3-x)]\)) to get: \(\ln [2]\)

Final Evaluation of the Limit

The limit is a constant, \(\ln [2]\), as \(x\) approaches 2 from the left. The function was continuous and defined at \(x = 2\), so the limit from the left is just the value of the function at \(x = 2\) itself.