Questions: Prove the following limits do not exist by finding both the left-hand and right-hand limit (a) lim (x → 3) ((x^2 + 2x - 3) / (x - 3))

Transcript text: Prove the following limits do not exist by finding both the left-hand and right-hand lim (a) $\lim _{\substack{x \rightarrow 3}}\left(\frac{x^{2}+2 x-3}{x-3}\right)$

Solution

Solution Steps

To determine if the limit exists, we need to evaluate the left-hand limit and the right-hand limit as $ x $ approaches 3. If these two limits are not equal, the limit does not exist.

Step 1: Define the Function

We are given the function: \[ f(x) = \frac{x^2 + 2x - 3}{x - 3} \]

Step 2: Simplify the Function

First, we factor the numerator: \[ x^2 + 2x - 3 = (x - 1)(x + 3) \] Thus, the function becomes: \[ f(x) = \frac{(x - 1)(x + 3)}{x - 3} \]

Step 3: Evaluate the Left-Hand Limit

We evaluate the left-hand limit as $ x $ approaches 3 from the left ($ x \to 3^- $): \[ \lim_{{x \to 3^-}} \frac{(x - 1)(x + 3)}{x - 3} = -\infty \]

Step 4: Evaluate the Right-Hand Limit

We evaluate the right-hand limit as $ x $ approaches 3 from the right ($ x \to 3^+ $): \[ \lim_{{x \to 3^+}} \frac{(x - 1)(x + 3)}{x - 3} = \infty \]

Step 5: Compare the Limits

Since the left-hand limit ($ -\infty $) and the right-hand limit ($ \infty $) are not equal, the limit does not exist.