Can we find out vertical asymptotes by finding the limit of a function y=f(x)/g(x) when y approaches infinity?
A vertical asymptotes occurs when y approaches infinity (like a horizontal asymptote occur when x approaches infinity). So, can you apply a limit as y approaches infinity, on any function, especially rational functions, to find the vertical asymptotes?
Why is it that vertical asymptotes are defined as the 'the values x can't equal'? How is the related to the limit as y approaches infinity?
Please consider that I am a high school student studying high level maths, so answer with explanations that you think I will understand.
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$\begingroup$Dividing by really small numbers yields really big numbers. Consider for instance the reciprocal function $1/x$. As you plug in numbers for $x$ that are closer and closer to $0$, the outputs of $1/x$ will get bigger and bigger in size (possibly negative, if you're plugging in negative numbers). This occurs with all rational functions $f(x)/g(x)$: as $x$ approaches a value in which $g(x)$ is $0$ but $f(x)$ isn't, you get division by smaller and smaller numbers yielding bigger and bigger outputs.
The $y$ coordinate of a point on the graph is the output of the function applied to the $x$ coordinate, so if the outputs are getting bigger in size, graphically that means the points on the graph are going way up or way down.
$\endgroup$ 3 $\begingroup$If you can find the inverse of the function and hence write it in the form $x=f^{-1}(y)$ then you can find vertical asymptotes by finding the limits:
$$\lim_{y\to\infty}f^{-1}(y)$$
and
$$\lim_{y\to-\infty}f^{-1}(y)$$
This is exactly the same as what you are doing with $x$ to find horizontal asymptotes.
However depending on your function you may not be able to find an inverse function to do this with.
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