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1-2 defining Limits and Limit Notation
Randy Eppert
Created on August 8, 2023
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Transcript
1-2: Defining Limits and Using Limit Notation
Learning Targets
- Represent a limit using correct notation
- Interpret a limit expressed in analytic notation
Start
The Perfect Cup of Coffee
1. A coffee connoisseur believes that the perfect temperature for a cup of coffee is 146°F. He use a thermometer to watch the temperature of his coffee for many years, and knows now that it takes exactly 349 seconds after the Keurig finishes for the coffee to cool to 146°.
The Perfect Cup of Coffee
Predict what you think the temperature of the coffee will be at each of these times.
349.001 seconds
348 seconds
147°
145.999°
146.5°
145.9°
348.5 seconds
349.1 seconds
348.9 seconds
349.5 seconds
146.01°
145.5°
350 seconds
348.999 seconds
146.0001°
145°
The Perfect Cup of Coffee
2. The limit of a function is the value that the function is approaching as the variable gets closer to a particular number. Here the temperature of the coffee is a function of time. We will call this function C(t). The value C(t) is approaching 146 as the t gets closer to 349 seconds. This is expressed using the notation:
The Perfect Cup of Coffee
Express the following ideas using the correct limit notation:
The Perfect Cup of Coffee
Important Ideas
- Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be arbitrarily close to R by taking x sufficiently close to c (but not equal to c).
- In other words, we can find a time close to 349 seconds where the temperature of the coffee is as close to 146° as we want. For example we could find a time near 349 seconds where the temperature of the coffee was within 0.01° of 146°.
- If the limit exists and is a real number, then the common notation is
The End