The Limit of a Function
The Limit of a Function
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 \(\lim_{x \to a} f(x) = L\text{,}\) as “the limit of \(f\) as \(x\) approaches \(a\) is \(L\text{,}\)” which means that we can make the value of \(f(x)\) as close to \(L\) as we want by taking \(x\) sufficiently close (but not equal) to \(a\text{.}\)
To find \(\lim_{x \to a} f(x)\) for a given value of \(a\) and a known function \(f\text{,}\) we can estimate this value from the graph of \(f\text{,}\) or we can make a table of function values for \(x\)-values that are closer and closer to \(a\text{.}\) 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 \(f\) change as \(x \to a\text{.}\)
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