Deeper Insight into the concept of the limit of functions

[00:00:00]Think of a limit as answering one simple question.

[00:00:03]What value is a function approaching as the input gets closer and closer to a particular point?

[00:00:09]The key word is approaching.

[00:00:11]We are not concerned with the value exactly at the point at first. We are interested in the trend of the function from nearby values.

[00:00:20]In the first solution, the function had different formulas on the left and right sides of a point.

[00:00:26]To determine whether the limit exists, we examined the left-hand limit and the right-hand limit separately.

[00:00:32]The left-hand limit tells us what happens when we approach the point from values smaller than the point.

[00:00:38]The right-hand limit tells us what happens when we approach from values larger than the point.

[00:00:44]A useful analogy is to imagine two students walking toward the same classroom from opposite ends of a corridor.

[00:00:52]If both students arrive at the same classroom, then there is agreement. In the language of limits, the left-hand limit and right-hand limit are equal.

[00:01:01]When this happens, the limit exists and equals that common value.

[00:01:05]However, the actual value of the function at the point may be different.

[00:01:09]This is one of the most important ideas in calculus.

[00:01:12]A limit depends on nearby values, while a function value depends on what the function is specifically assigned at that point.

[00:01:21]Therefore, it is possible for a limit to exist even when the function value is different. When that happens, the graph has a discontinuity, often called a removable discontinuity, because the mismatch can be fixed by redefining the function value.

[00:01:35]The second solution explains the general conditions under which a limit exists at a point x equals a.

[00:01:42]These conditions are not separate facts to memorize. They form a logical test.

[00:01:47]First, the left-hand limit must exist.

[00:01:50]This means the function should approach a definite value from the left side of a.

[00:01:54]Second, the right-hand limit must exist.

[00:01:57]This means the function should also approach a definite value from the right side of A.

[00:02:02]Third, and most importantly, these two values must be equal.

[00:02:07]If even one of these conditions fails, the limit does not exist.

[00:02:12]For example, if the left-hand limit is three and the right-hand limit is five, there is no single value that the function approaches.

[00:02:19]It would be like one student arriving at room three and another arriving at room five.

[00:02:24]There is no common destination.

[00:02:26]Notice that the conditions say nothing about the actual value f of A.

[00:02:30]This is deliberate.

[00:02:32]The existence of a limit depends entirely on the behavior of the function near the point, not necessarily at the point itself.

[00:02:40]Many students confuse limits with function values, but they are different concepts.

[00:02:45]The big idea connecting both solutions is this. Whenever you are asked whether a limit exists, always check the left-hand limit, check the right-hand limit, and compare them.

[00:02:56]If they exist and are equal, the limit exists.

[00:03:00]After that, compare the limit with the actual function value.

[00:03:04]If they are equal, the function is continuous at that point. If they are different, the limit still exists, but the function is not continuous there.

[00:03:13]This simple procedure will solve most introductory limit problems correctly and confidently.