Question 507121: Use a graph of f or some other method to determine what, if any, value to assign to f(a) to make f continuous at x = a. (If there is no such value, enter NONE.)
f(x)=X^2+13x+42/x+7 ; a=-7
Answer by tinbar(133)   (Show Source):
You can put this solution on YOUR website! To make it continuous, you simply need to guarantee the limit of the function at any given point reaches the same value regardless of approaching from the left or the right.
Now your given f(x) is continuous and well defined except for x=-7, since the denominator goes to 0 and the function goes to infinity. If you take the limit though as a approaches -7 though, you can factor out an (x+7) from the top and cancel it with the bottom. X^2+13x+42/x+7 = (x+6)(x+7)/x+7 = x+6. So now we sub x = -7, and we get -7+6 = -1. Therefore, let f(a) = -1.
If this is not clear, go to http://www.coolmath.com/graphit/ and after the calculator loads up, enter (x^2+13x+42)/(x+7) into the calculator and hit the 'eval' button (first column, fourth row). When it plots the graph, you can see there's a 'hole' at x=-7, and that if the point (-7,-1) is dotted in, the line becomes continuous.
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