Section 1-5
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The document discusses the concept of limits in calculus, including the formal definition of a limit as the value a function approaches as the input gets arbitrarily close to a given number. It provides examples of evaluating limits graphically and algebraically, such as determining the limit of a function as x approaches 3 is 6, even though the function is undefined at x=3 due to a removable discontinuity. The document also discusses using graphs to identify different types of discontinuities and limits.
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- 2. The function is undefined at x = 3, and we say that it has a removable discontinuity . Usually this is indicated with an open circle on the graph (even though your calculator can’t do that). If we factor the numerator of the function, we can see what is happening… It seems as if the point (3,6) should be on the graph, but it can’t be because of the original definition of the function. We say that 6 is the limit of the function as x approaches 3.
- 3. The Formal Definition of a Limit L is the limit of f ( x ) as x approaches c if and only if for any positive number epsilon, no matter how small, there is a positive number delta such that if x is within delta units of c , (but not equal to c ), then f ( x ) is within epsilon units of L . Huh?
- 4. The Formal Definition of a Limit L is the limit of f ( x ) as x approaches c if and only if for any positive number epsilon, no matter how small, there is a positive number delta such that if x is within delta units of c , (but not equal to c ), then f ( x ) is within epsilon units of L . In our previous example, 6 was the limit of f ( x ) as x approaches 3. According to the definition above, we can make f (x) as close as we want to 6 by choosing x to be close enough to 3.
- 5. Limits from a Graph Limits can usually be seen readily on a graph: What is the limit of the function as x approaches 6? as x approaches -4? as x approaches -2?