Limits from graphs | Limits and continuity | AP Calculus AB | Khan Academy

This paragraph introduces the concept of graphical limits in the context of a function represented by the graph of y = f(x). The instructor begins by posing a question about the limit as x approaches 6, encouraging viewers to attempt to solve it before moving forward. The analysis involves observing the graph to determine the behavior of f(x) as x gets closer to 6 from both the left and right sides. The conclusion is that the limit appears to be 1, as the function values approach the same value from both directions. The discussion then shifts to the limit as x approaches 4, noting that even though the function is not defined at x = 4, the limit can still be determined from the graph. In this case, the limit is found to be 3, as the function values approach the same value from both the left and right sides of x = 4. The third limit considered is as x approaches 2. Here, the function is defined at x = 2, but the limit does not exist because the function values approach different values from the left and right sides. This example illustrates that a function can be defined at a point but still not have a limit as x approaches that point. The paragraph concludes with a brief mention of looking at more cases to understand graphical limits, setting up the stage for further analysis in subsequent paragraphs.

This paragraph continues the exploration of graphical limits with a focus on the function g(x). The first limit examined is as x approaches 5. The instructor guides the viewer to observe the graph and estimate the limit by drawing a straight line to represent the function's behavior as it nears the value of x = 5. The visual estimation suggests a limit of approximately 0.4, although the exact value cannot be determined without further calculation. The second limit discussed is as x approaches 7. The analysis involves observing the function's behavior from values less than 7, such as 6.9, 6.99, and 6.999. The function appears to approach the value of 2 from both the left and right sides of x = 7. However, it is noted that even though the function is defined at x = 7, the limit value is different from the function's value at that point. The final limit considered in this paragraph is as x approaches 1. The graph indicates that the function is unbounded as it approaches this value, with function values moving towards infinity from both the left and right sides. The instructor emphasizes the importance of being formal in the context of limits, stating that the limit does not exist because the function is unbounded at this point. The paragraph ends with a summary of the key points discussed, reinforcing the concept that a function can be defined at a point but not have a limit, and that limits can exist even if the function is not defined at a particular x-value.