The Limit of a Function

Consider the functionWe can perform long division to obtainbut this is not equal toThe reason is that ifbecomeswhich is not defined, but

We can sketch the graphs ofandThis is done below.

The graph ofis not defined atand this is indicated by an empty circle at the point on the graph whereWe can however define the limit of the functionastends to 1.

The graphitself is not defined atbut it does not need to be. As long astends to some limit, we can define the limit ofatby considering

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