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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, consider the behavior of \(\frac{2}{{2 + 3x}}\) as \(2 + 3x\) approaches zero from the positive side. Since the numerator is positive, as the denominator gets smaller and smaller (approaches zero from the positive side), the entire fraction becomes very large: \[ \lim_{{x \to -\frac{2}{3}^-}} \frac{2}{{2 + 3x}} = +\infty \] #### Step 3: Evaluate the right-hand limit (RHL) The right-hand limit is when \(x\) approaches \(-\frac{2}{3}\) from the right 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 right, \(3x\) approaches \(-2\) from the right. Therefore, \(2 + 3x\) becomes a small negative number approaching zero from the negative side. 3. Now, consider the behavior of \(\frac{2}{{2 + 3x}}\) as \(2 + 3x\) approaches zero from the negative side. Since the numerator is positive, as the denominator gets smaller and smaller (approaches zero from the negative side), the entire fraction becomes very large but negative: \[ \lim_{{x \to -\frac{2}{3}^+}} \frac{2}{{2 + 3x}} = -\infty \] #### Step 4: Compare the left-hand and right-hand limits - The left-hand limit (LHL) as \(x \to -\frac{2}{3}^-\) is \(+\infty\). - The right-hand limit (RHL) as \(x \to -\frac{2}{3}^+\) is \(-\infty\). Since the LHL and RHL are not equal (one is \(+\infty\) and the other is \(-\infty\)), the limit does not exist. ### Conclusion: The limit \(\lim_{{x \to -\frac{2}{3}}} \frac{2}{{2 + 3x}}\) does not exist.