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Explain why the graph of the rational function f(x)= (-1)/(X^2+4) has no vertical asymptotes

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2 Answers

If we set the denominator equal to zero and solve for x, we won't get a real solution.  Therefore, the graph does not have any vertical asymptotes.  Therefore, the function is continuous.
 
x2 + 4 = 0
 
x2 = -4
 
x = 2i    and   x = -2i
 
These solutions are complex. 
In order to have vertical asymptotes, the denominator x2 + 4 would have to be zero for some value of x.  But x2 + 4 > 0 for all real values of x.  (There are complex values of x that make x2 + 4 = 0, but I assume we are concerned only with real numbers here.)