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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 → a f ( x ) = L , as “the limit of f as x approaches a is L , ” 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 .

To find lim x → a f ( x ) for a given value of a and a known function f , we can estimate this value from the graph of f , or we can make a table of function values for x -values that are closer and closer to a . 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 → a .

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

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