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Consider the function
Notice that is undefined when
However, we still may want to analyze the behavior of
around
The mathematical tool used to do this sort of analysis is called a limit.
To see how this works graphically, shown below is the graph of
Notice that there is a hole in the graph at the point indicating that the graph of
is a line, but excludes the point
Since is undefined when
we analyze the behavior of
by using limits.
That is, we want to evaluate or more specifically,
By examining the graph, it appears that as x gets closer and closer to 1, gets closer and closer to 2. Thus, we can write
We can use the information from a graph to evaluate a limit.
EXAMPLE
Consider the graph of some functionStatement | Description |
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As x gets closer to 0, ![]() |
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As x gets closer to 1, ![]() |
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As x gets closer to 2 from the left (values smaller than 2), ![]() ![]() Since ![]() |
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As x gets closer to 3, ![]() ![]() ![]() |
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As x gets closer to 4, ![]() |
Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 1 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.