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# limits - Chapter 3 Limits and Continuity Chapter intro here...

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Chapter 3 – Limits and Continuity Chapter intro here….. Limits found Graphically A limit is the idea of looking at what happens to a function as you approach particular values of x . Left-hand and right-hand limits are the idea of looking at what happens to a function as you approach a particular value of x , from a particular direction . The limit of f(x) as x approaches the value of a from the left is written ( ) lim x a f x - and the limit of f(x) as x approaches the value of a from the right is written ( ) lim x a f x + Let’s explore these ideas with the graph of f(x) in Figure 3.1 below. Figure 3.1 Looking at f(x) when x = -2 , you notice there is a “break” in the function. However, if you approach x = -2 “from the left” (Figure 3.2a) you can see that the function values are getting closer and closer to 1. On the other hand, if we approach x = -2 “from the right” (Figure 3.2b) you can see that the function values are getting closer and closer to 3.

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Figure 3.2a Figure 3.2b Looking at f(x) when x = 1 , you notice there is a hole in the function. If we approach f(x) from the left or from the right (Figure 3.3), you can see that the function values are getting closer and closer to 2. Figure 3.3 Therefore, the following statements are true. 2 ( ) 1 lim x f x - →- = 2 ( ) 3 lim x f x + →- = 1 ( ) 2 lim x f x - = 1 ( ) 2 lim x f x + =
Example 1 Using the given graph of g(x) , find the following left- and right-hand limits. a. 0 ( ) lim x g x - b. 0 ( ) lim x g x + c. 1 ( ) lim x g x - d. 1 ( ) lim x g x + Solution a. This asks us to look at the graph of g(x) as x approaches 0 from the left. You can see that the function values are getting closer and closer to -1 . So, 0 ( ) 1 lim x g x - = - b. This asks us to look at the graph of g(x) as x approaches 0 from the right. You can see that the function values are getting closer and closer to -1 . So, 0 ( ) 1 lim x g x + = -

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c. This asks us to look at the graph of g(x) as x approaches 1 from the left. You can see that the function values are getting closer and closer to -2 . So, 1 ( ) 2 lim x g x - = - d. This asks us to look at the graph of g(x) as x approaches 1 from the right. You can see that the function values are getting closer and closer to -2 . So, 1 ( ) 2 lim x g x + = - Notice that in the solutions to parts (c) and (d) above, the function value g(1)=1 does not play a role in determining the values of the limits. A limit is strictly the behavior of a function “near” a point. Example 2 Using the graph of h(x) below, find the following left- and right-hand limits. a. 4 ( ) lim x h x - b. 4 ( ) lim x h x +
Solution a. Looking at the graph of h(x) , as x approaches 4 from the left, you can see that the function values keep getting more and more negative, without end. Thus, we say that the function values approach negative infinity, written 4 ( ) lim x h x - = -∞ GRAPH? b. Looking at the graph of h(x) , as x approaches 4 from the right, you can see that the function values keep getting more and more positive without end. Thus, we say that the function values approach positive infinity, written 4 ( ) lim x h x + = ∞ GRAPH?

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limits - Chapter 3 Limits and Continuity Chapter intro here...

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