Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
The Limit of a Function- L1
Objectives
- Using correct notation, describe the limit of a function.
- Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
- Use a graph to estimate the limit of a function or to identify when the limit does not exist.
- Define one-sided limits and provide examples.
- Explain the relationship between one-sided and two-sided limits.
- Using correct notation, describe an infinite limit.
- Define a vertical asymptote.
Summary
Limits enable us to examine trends in function behavior near a specific point. In particular, taking a limit at a given point asks if the function values nearby tend to approach a particular fixed value.
We read as “the limit of as approaches is ” which means that we can make the value of as close to as we want by taking sufficiently close (but not equal) to
To find for a given value of and a known function we can estimate this value from the graph of or we can make a table of function values for -values that are closer and closer to If we want the exact value of the limit, we can work with the function algebraically to understand how different parts of the formula for change as
We find the instantaneous velocity of a moving object at a fixed time by taking the limit of average velocities of the object over shorter and shorter time intervals containing the time of interest.
See the Desmos demonstration on Average Velocity & Secant Lines