Question 332814
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A rational function:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ r(x)\ =\ \frac{f(x)}{g(x)}]


has a vertical asymptote at any value of *[tex \Large x] such that *[tex \Large g(x)\ =\ 0], so long as any factors common to both *[tex \Large f(x)] and *[tex \Large g(x)] have been eliminated.


If *[tex \Large g(\alpha)\ =\ 0], then the line represented by the equation *[tex \Large x\ =\ \alpha] is a vertical asymptote.


For your given function, *[tex \Large f(x)\ =\ x\ +\ 3] and *[tex \Large g(x)\ =\ x(x\ +\ 1)].  Clearly these two functions have no factors in common, so simply set:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ g(x)\ =\ x(x\ +\ 1)\ =\ 0]


And solve for all values of *[tex \Large x].  Each of these values is an *[tex \Large \alpha] value that completes an equation of a vertical asymptote, *[tex \Large x\ =\ \alpha] for your rational function.


Compare your results to the graph below:


{{{drawing(
500, 500, -5, 5, -12, 5,
grid(1),
graph(
500, 500, -5, 5, -12, 5,
(x+3)/(x^2+x)
))}}} 


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
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