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While working out some examples I'm trying to solve, I stumbled on a question that asks to find the asymptotes of the following function:

$$y = \sqrt{x^2 + 3}$$

For rational functions I was thought to perform long division for horizontal/oblique asymptotes which in this case there are 2 oblique.

How to I find these asymptotes without performing the limits method since I have no idea how to do it and we weren't thought that method in class.

Thanks

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Probably what the instructor wants you to realize is that when $x$ is very large in magnitude (either positive or negative), your function is approximately $\sqrt{x^2}=|x|$.

If you graph your function and $|x|$, you will see the root function approaches the absolute value function in the long term.

This is analogous to what happens when rational functions have "oblique" asymptotes, as they are called in high school. (I highly doubt you'll ever hear anyone outside of high school refer to such things.) For example, the function $$f(x)=\frac{x^3}{x^2-1}$$ is approximately $x^3/x^2=x$ when $x$ is large in magnitude, so in the long run this rational function approaches the line $y=x$.

The important idea here is to think approximately.

Another approach is to view your function as the upper branch of a hyperbola, but this won't make sense if you haven't studied conic sections.

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We can re-write the equation as $$ y^2 - x^2 = 3 $$ which factors as $$(y - x)(y + x) = 3$$

We can substitute $u = y - x$ and $v = y + x$, and the resulting equation is $$ uv = 3 $$ which has asymptotes $u = 0$ and $v = 0$.

Substituting the old variables back in tells us that the asymptotes are $y = -x$ and $y = x$.

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