- Observe that as \(x \to 0\), the function \(f(x)\) approaches a finite value from both sides, indicating a hole at \(x=0\).
- Observe that as \(x \to 4\), the function \(f(x)\) values increase or decrease without bound, indicating a vertical asymptote at \(x=4\).
- Conclude that the correct statement is: The function has a hole when \(x=0\) and a vertical asymptote when \(x=4\).
- \(\boxed{\text{The function has a hole when } x=0 \text{ and a vertical asymptote when } x=4.}\)
Explanation
- Analyzing Behavior at \(x=0\) and \(x=4\)
To determine the nature of the function \(f(x)\) at the points \(x=0\) and \(x=4\), we examine the behavior of the function values as \(x\) approaches these points from the left and right sides. Analyzing the behavior near a point helps distinguish between a removable discontinuity (a hole) and an infinite discontinuity (a vertical asymptote).
- Behavior at \(x=0\)
At \(x=0\), looking at values close to 0:
- As \(x\) approaches \(0\) from the left (\(x = -0.2, -0.1\)), \(f(x)\) takes values \(-0.238\) and \(-0.244\). These values are close to \(-0.25\).
- As \(x\) approaches \(0\) from the right (\(x = 0.1, 0.2\)), \(f(x)\) takes values \(-0.256\) and \(-0.263\). These values are also close to \(-0.25\).
Since the limit of \(f(x)\) as \(x \to 0\) exists and is a finite number, but the function is explicitly undefined at \(x=0\), there is a removable discontinuity, or a hole, at \(x=0\).
- Behavior at \(x=4\)
At \(x=4\), looking at values close to 4:
- As \(x\) approaches \(4\) from the left (\(x = 3.9, 3.99\)), \(f(x)\) takes values \(-10, -100\). The values are decreasing without bound, approaching \(-\infty\).
- As \(x\) approaches \(4\) from the right (\(x = 4.01, 4.1\)), \(f(x)\) takes values \(100, 10\). The values are increasing without bound, approaching \(+\infty\).
Because the function values grow infinitely large in magnitude as \(x\) approaches \(4\), the line \(x=4\) represents a vertical asymptote.
- Conclusion and Final Answer
Comparing these findings to the available choices:
- A hole occurs at \(x=0\) because the limit is finite.
- A vertical asymptote occurs at \(x=4\) because the function approaches \(\pm\infty\).
Therefore, the correct description is that the function has a hole when \(x=0\) and a vertical asymptote when \(x=4\).
Examples
In civil engineering, specifically when designing the structural integrity of bridges or support beams, engineers model stress distributions across various points. If a stress model shows a finite limit at a specific connection point despite the point being marked 'undefined' due to simplified load assumptions, it often indicates a 'removable' structural singularity that can be smoothed over. Conversely, if values approach infinity, it signifies a 'vertical asymptote' or a critical failure point where stress becomes unmanageable, requiring immediate design intervention to prevent collapse.