1-B Investigating Limits
Instructions
Use the input boxes on the left to define a function by entering the numerator and denominator functions. Note the values excluded from the domain of the function. The checkbox will show/hide the graph of the function.
- Use the slider tool or input box to move the point x near to the excluded values. Notice the point leaves a trace of the graph of the function. (Click "Clear Trace" button to erase it.)
- Observe the behavior of the function near the excluded values. Note: the function values are displayed in function notation under the input box for x.
1-B The Limit Concept
Although almost all of Calculus relies on the limit concept (from a logical, theoretical standpoint), it was actually the last major concept of Calculus to be developed formally. This means that for around 200 years (mid 1600s - mid 1800s) people were using Calculus without a formal approach to the limit concept. This is why I feel justified in choosing to focus on the informal, intuitive conception of limits.
We know that functions often have "domain issues" (e.g., an input that doesn't produce an output). For example, rational functions are undefined where the denominator is equal to 0 because division by 0 does not produce a numerical value for the output. When we find such excluded values (i.e., values not in the domain of the function), a natural question to ask is: "What does the function do (i.e., how does it behave) when the input is near the excluded value?"
An informal definition: We say that a function has a limit as approaches if we observe to become progressively closer to whenever gets progressively closer to . Notice that the limit is a numerical value (not a process). The notation for this is: .