Ch. 8: Limits and Continuity EXAMPLE 17 .

It seems like you're referring to a specific example from a textbook or resource on limits and continuity, but I don't have access to that specific content. However, I can help explain concepts related to limits and continuity or provide a general example. ### Example of Limits and Continuity **Limit Example:** Let's consider the function \( f(x) = \frac{x^2 - 1}{x - 1} \). To find the limit as \( x \) approaches 1, we can first try to substitute \( x = 1 \): \[ f(1) = \frac{1^2 - 1}{1 - 1} = \frac{0}{0} \] This is an indeterminate form, so we need to simplify the function. We can factor the numerator: \[ f(x) = \frac{(x - 1)(x + 1)}{x - 1} \] For \( x \neq 1 \), we can cancel \( x - 1 \): \[ f(x) = x + 1 \quad \text{for } x \neq 1 \] Now we can find the limit as \( x \) approaches 1: \[ \lim_{x \to 1} f(x) = \lim_{x \to 1} (x + 1) = 1 + 1 = 2 \] Thus, the limit of \( f(x) \) as \( x \) approaches 1 is 2. **Continuity Example:** A function is continuous at a point \( c \) if the following three conditions are met: 1. \( f(c) \) is defined. 2. \( \lim_{x \to c} f(x) \) exists. 3. \( \lim_{x \to c} f(x) = f(c) \). Using the previous function \( f(x) \): 1. \( f(1) \) is not defined because it results in \( \frac{0}{0} \). 2. The limit \( \lim_{x \to 1} f(x) = 2 \) exists. 3. Since \( f(1) \) is not defined, the function is not continuous at \( x = 1 \). If you have a specific example or problem in mind from your text, please provide more details, and I can help you with that!