Explain the meaning of each of the following. (a) \( \lim _{x \rightarrow-3} f(x)=\infty \) \( f(-3)=\infty \) The values of \( f(x) \) can be made arbitrarily close to -3 by taking \( x \) sufficiently large. The values of \( f(x) \) can be made arbitrarily close to 0 by taking \( x \) sufficiently close to (but not equal to) -3 . The values of \( f(x) \) can be made arbitrarily large by taking \( x \) sufficiently close to (but not equal to) -3 . (b) \( \lim _{x \rightarrow 2^{+}} f(x)=-\infty \) The values of \( f(x)=-\infty \) The values of \( f(x) \) can be made negative with arbitrarily large absolute values by taking \( x \) sufficiently close but greater than, 2 . As \( x \) approaches \( 2, f(x) \) approaches \( -\infty \).

Let's break down the meanings of each statement in parts (a) and (b) regarding limits and function behavior. ### Part (a) 1. **\( \lim _{x \rightarrow -3} f(x) = \infty \)**: - This means that as \( x \) approaches -3 from either side (left or right), the values of the function \( f(x) \) increase without bound. In simpler terms, \( f(x) \) becomes larger and larger as \( x \) gets closer to -3. 2. **\( f(-3) = \infty \)**: - This statement is somewhat misleading. In standard mathematical terms, we typically do not assign a value of infinity to a function at a specific point. Instead, it indicates that the function does not have a finite value at \( x = -3 \). Instead, it suggests that \( f(x) \) diverges to infinity as \( x \) approaches -3. 3. **The values of \( f(x) \) can be made arbitrarily close to -3 by taking \( x \) sufficiently large**: - This statement is incorrect in the context of limits. The values of \( f(x) \) do not approach -3; rather, it is \( x \) that approaches -3. The correct interpretation would be that \( x \) can be made arbitrarily close to -3, but this does not relate to the behavior of \( f(x) \). 4. **The values of \( f(x) \) can be made arbitrarily close to 0 by taking \( x \) sufficiently close to (but not equal to) -3**: - This statement is also incorrect. As \( x \) approaches -3, \( f(x) \) does not approach 0; instead, it approaches infinity. 5. **The values of \( f(x) \) can be made arbitrarily large by taking \( x \) sufficiently close to (but not equal to) -3**: - This statement is correct. It reinforces the idea that as \( x \) gets closer to -3 (but not equal to -3), the values of \( f(x) \) increase without bound, meaning they can be made arbitrarily large. ### Part (b) 1. **\( \lim _{x \rightarrow 2^{+}} f(x) = -\infty \)**: - This means that as \( x \) approaches 2 from the right (values greater than 2), the values of the function \( f(x) \) decrease without bound. In other words, \( f(x) \) becomes more and more negative as \( x \) gets closer to 2 from the right. 2. **The values of \( f(x) = -\infty \)**: - Similar to part (a), this statement is misleading. It suggests that at \( x = 2 \), the function takes the value of negative infinity, which is not a valid assignment. Instead, it indicates that \( f(x) \) diverges to negative infinity as \( x \) approaches 2. 3. **The values of \( f(x) \) can be made negative with arbitrarily large absolute values by taking \( x \) sufficiently close but greater than 2**: - This statement is correct. It means that as \( x \) gets closer to 2 from the right, the values of \( f(x) \) can be made very large in the negative direction (i.e., very negative). 4. **As \( x \) approaches \( 2, f(x) \) approaches \( -\infty \)**: - This statement is a restatement of the limit definition. It confirms that as \( x \) gets closer to 2 from the right, the function \( f(x) \) approaches negative infinity. In summary, parts (a) and (b) describe the behavior of the function \( f(x) \) near specific points, highlighting how the function diverges to infinity or negative infinity as \( x \) approaches those points.