Limits from a Graph
The Limit Concept (One-Sided Limits)
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.
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: .
In practice, we are typically concerned with determining whether a limit exists, and estimating its value, if it does exist. We do this by letting x approach c either from the left () or from the right () and observing the corresponding function values f(x). If the one-sided limits exist and are equal to the same number L, then the two-sided limit also exists and is equal to L.
Instructions
Use the input boxes to enter formulas for the left (red) and right (blue) pieces of a piecewise defined function. (You can use the same function formula in both input boxes to get a "normal" graph.)
- Use the input box for c to change the point where you will investigate the limits.
- Use the input box for x to set a starting point for your investigation.
- Use the x \to c button to have x move closer to c.
- Use the Trace On / Off buttons to leave a trace of the function values on the y-axis to observe whether they approach a particular value. Use Clear Trace to remove all trace points.
- When the function values get "close enough" to the limit value, the one-sided limit notation will be displayed on the bottom-left of the screen.