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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 …).
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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:
