This is not the case when f(c) does not exist, though. On the right are some examples of a function not existing at a certain point; these should be identifiable based on Algebra II knowledge.
The distinction between a limit and a value is extremely important. An analogy would be driving a car towards a destination. The car is driving towards its destination; when it reaches its destination, it is turned off. The value - f(c) - is the location of the car when it is turned off. The limit, on the other hand, is the location of the car less than a split second before the ignition is turned off - it is where the car has been heading, but is just about to have reached there. When a function does not exist, the ignition never stops - the limit exists, but a solution (a value) never materializes. |
In the first case, the function does not exist at x=3, because if one "plugs in" x=3, the denominator becomes zero, which is not possible in mathematics.
In the second case, the function does not exist when x < -1, because any number under -1 makes the number under the square root radical. No real number exists that is the square root of a negative number; therefore, the function does not exist there. In the third case, the function is positive exponential; thus, the function cannot palpably exist below the x-axis (i.e. y = 0 is an asymptote). |