What are the coordinates of the focus of the conic section shown below?
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Your formula for this is 
and 
. Get everything on one side of the equals sign, set it equal to 0 and factor. When you do this you get (x-3)(x+27). The Zero Product Property rule tells us that either x-3 = 0 or x+27 = 0 and that x = 3 and -27. The only thing in math that will NEVER be negative besides time is distance/length, therefore, x cannot be 27 and has to be 3.
If we evaluate the function at infinity, we can immediately see that:
Therefore, we must perform an algebraic manipulation in order to get rid of the indeterminacy.
We can solve this limit in two ways.
<h3>Way 1:</h3>By comparison of infinities:
We first expand the binomial squared, so we get
Note that in the numerator we get x⁴ while in the denominator we get x³ as the highest degree terms. Therefore, the degree of the numerator is greater and the limit will be \infty. Recall that when the degree of the numerator is greater, then the limit is \infty if the terms of greater degree have the same sign.
<h3>Way 2</h3>Dividing numerator and denominator by the term of highest degree:
Note that, in general, 1/0 is an indeterminate form. However, we are computing a limit when x →∞, and both the numerator and denominator are positive as x grows, so we can conclude that the limit will be ∞.