Limit notation is written so that we can identify the end behavior of a graph. The limit notation allows us to have a visual or guide to how the graphs would will look. In a horizontal asymptote the limit notation serves as a type of guide for the graphs. It tells you where the horizontal y axis is just as a fact of how the behaviors are when approaching positive and negative infinity. More specifically, it lets you know if the graph is heading towards the number given.
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