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Limits: An Informal View

A function f can have a specific value for an input. For example, if f (x) = 0.5x then f (5) = 2.5. For continuous functions like this one, as x approaches 5, the output value of f approaches 2.5. For some other functions, however, we can extend this concept to cases where the function is not continuous or even has no value. Informally, we say that if f (x) approaches L as x approaches c, then L is called the limit. We write this mathematically as Limit x->c f(x) = L.

Try the following:

  1. The first graph show the line used in the example above. Is the limit L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Move the x slider so that x gets closer and closer to 1. For this example, the limit does equal 0.5. In fact, f (1) = 0.5 in this case, but as we will see, this does not need to be the case.

  2. Select the second example. This is just like the first case, except that one point has moved. Is the limit still L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Move the x slider so that x gets closer and closer to 1. For this example, the limit still equals 0.5. As x gets closer and closer to c = 1, f (x) gets closer and closer to L = 0.5. The limit is not 1.5, even though that is the value of the function at x = 1. Instead, the limit is the output value that is approached as the input value approaches 1. For functions that are continuous around the point of interest, as in the first example, it is easy to find the limit; you just evaluate the function at the specific input, because the output value it approaches is the same as the function's output value at that point. But for cases like this second example, there still is a limit but it happens to be different than the value of the function.

  3. Select the third example. This is like the previous two cases, but there is now a point missing. Is the limit still L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Move the x slider so that x gets closer and closer to 1. For this example, the limit still equals 0.5. As x approaches c = 1, f (x) approaches L = 0.5. Even though the function has no value at all at x = 1, there is still a valid limit there, as the function does approach a specific output value.

  4. Select the fourth example. This is a more complex function, but this example is similar to the previous one with a missing point. What is the limit when c = 0 ? In other words, what value does f (x) approach as x approaches 0? Move the x slider so that x gets closer and closer to 0. For this example, the limit equals 1. As x gets closer and closer to c = 0, f (x) gets closer and closer to L = 1. Even though the function has no value at all at x = 0, there is still a valid limit there, as the function does approach a specific value. Mathematically, we would write lim x->0 sin(x)/x = 1.

Limits are part of the foundation of the theory of calculus. They can also make the informal definition of continuity more formal.

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This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.

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