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LIMITS What is Calculus? What are Limits? Evaluating Limits –Graphically –Numerically –Analytically What is Continuity? Infinite Limits This presentation.

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1 LIMITS What is Calculus? What are Limits? Evaluating Limits –Graphically –Numerically –Analytically What is Continuity? Infinite Limits This presentation

2 Continuity at a point 1.f(c) is defined 2. exists 3. A function, f, is continuous at point, x = c, if there is no interruption of the graph of f at c. f is continuous at c, if three conditions are met:

3 Discontinuity at a Point f(c) is not defined If f is not continuous at c, then it is said to be discontinuous at c dne This can occur if any of the 3 conditions of continuity is not met: Examples 1 & 3 are removable discontinuities because f can be made continuous by appropriately defining or redefining f(c). [These are point discontinuities] Example 2 is an example of a non-removable discontinuity. [These occurs at “jumps” or at vertical asymptotes.]

4 Continuity on an Open Interval A function is continuous on an open interval, (a, b), if it is continuous at each point in the interval. A function that is continuous on the entire real line,, is everywhere continuous. Examples of such functions are all polynomials, and the sin(x) and cos(x) functions. Why is an open interval specified? Because part of the definition of continuity at a point involves finding the limit at that point. To do this, the point must be approachable on both sides. This would not be possible for points at the edges of a closed interval.

5 Continuity of a function

6 One-Sided Limits A one-sided limit means that x approaches c from one direction. The limit from the right means that x approaches c from values greater than c and is represented by: The limit from the left means that x approaches c from values less than c and is represented by: Important note: if and only if andRecall that

7 One-Sided Limits: An Example The greatest integer function: The limit as x approaches 0 from the left is The limit as x approaches 0 from the right is dne The limit as x approaches 0 does not exist, since the left and right sided limits are not the same. Since, the function is not continuous at x = 0. By similar reasoning, this function is not continuous at any integer. This function returns the greatest integer less than or equal to x.

8 Continuity & One-Sided Limits: Summary Three conditions must be met in order for a function to be continuous at a point, c –The function must be defined at c –The limit must be defined at c –The two values above must be the same A function can be continuous over an open interval. If that interval is the entire number line, then the function is everywhere continuous. Limits can be evaluated on one side of a point. However, the limit at that point does not exist unless both one-sided limits are the same.


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