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Let's solve the problem step by step. ### Problem: Prove that \(\lim_{{x \to -\frac{2}{3}}} \frac{2}{{2 + 3x}}\) does not exist. To prove that this limit does not exist, we need to analyze the behavior of the function \(\frac{2}{{2 + 3x}}\) as \(x\) approaches \(-\frac{2}{3}\) from both the left side and the right side. #### Step 1: Identify the expression inside the limit The function is given as: \[ f(x) = \frac{2}{{2 + 3x}} \] We need to evaluate the limit as \(x \to -\frac{2}{3}\). #### Step 2: Evaluate the left-hand limit (LHL) The left-hand limit is when \(x\) approaches \(-\frac{2}{3}\) from the left side (i.e., \(x \to -\frac{2}{3}^-\)). 1. Substitute \(x = -\frac{2}{3}\) into the expression \(2 + 3x\): \[ 2 + 3\left(-\frac{2}{3}\right) = 2 - 2 = 0 \] 2. As \(x\) approaches \(-\frac{2}{3}\) from the left, \(3x\) approaches \(-2\) from the left. Therefore, \(2 + 3x\) becomes a small positive number approaching zero from the positive side. 3. Now, con
