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