Understanding the Limit of a Function and Limit Laws in Calculus
This guide explores the concept of limits in calculus, defining what a limit is and how it behaves as a function approaches a specific value. We examine the significance of limits, including their role in determining the behavior of functions near a point rather than at the point itself. The document covers limit laws, special notations, and includes application examples such as the Sandwich Theorem. By understanding these principles, readers will gain insight into fundamental calculus concepts crucial for advanced mathematics.
Understanding the Limit of a Function and Limit Laws in Calculus
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Presentation Transcript
Math 180 2.2 – Limit of a Function and Limit Laws
Consider . What is ? ________________
Consider . What is ? ________________ undefined
Consider . What is ? ________________ undefined (Well, technically indeterminate…)
Now what happens to as gets close to 1? It looks like approaches _____ as approaches 1.
Now what happens to as gets close to 1? 2 It looks like approaches _____ as approaches 1.
Now what happens to as gets close to 1? 2 It looks like approaches _____ as approaches 1. Using special math notation, we can write the behavior like this:
Limit of a Function If can be as close to as we like by choosing -values close to a number (from both sides), then is the limit of as approaches . That is, (read: “the limit of as approaches is ”)
Note that with limits, we only care about the behavior of functions near, not at. So, in all three graphs below, .
Ex 1. What is ? What is ?
Ex 1. What is ? What is ? does not exist
Ex 1. What is ? What is ? does not exist does not exist
The Limit Laws Note: Below, and are constants (real numbers). In general, ___ and ___.
The Limit Laws Note: Below, and are constants (real numbers). In general, ___ and ___.
The Limit Laws Note: Below, and are constants (real numbers). In general, ___ and ___.
The Limit Laws If and both exist, then the following laws are true:
Ex 3. Find
The Sandwich Theorem Suppose that for all in some open interval containing , except possibly at itself. Suppose also that Then .
The Sandwich Theorem Note: The Sandwich Theorem has other names: Squeeze Theorem, Pinching Theorem, Two Policemen and a Drunk Theorem, etc.
Note: The Sandwich Theorem can be used to prove the following:
