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I have a function:

$$ y = \frac{20x^2 + 14x + 2}{7m+11xm-m\sqrt{37+70x+x^2}} $$

Note: In this case, m is simply some parameter

When I plot out this graph, it looks asymptotically linear.

How can I go about checking:

  1. If the function is, in fact, asymptotically linear
  2. If so, what is the asymptotic function's formula.
Mui
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  • You can take the limit of $y/x$ as $x \rightarrow \pm \infty$, and show that the limit is a constant. – Doug Aug 16 '22 at 17:14
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    You can see that $y/x\to 20/(10m)$ and that $y-2x/m$ also converges to a finite limit as $x\to\pm\infty$. I haven't compute the second. It looks like it also exists and it is zero, but check it. You shouldn't skip the second limit, since the first alone doesn't imply the existence of an asymptote (a linear one). If the second limit is $a$ then the line $y=2x/m + a$ is the asymptote in the direction that you computed the limits, $-\infty$ or $+\infty$. – plop Aug 16 '22 at 17:21

2 Answers2

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When I plot out this graph, it looks asymptotically linear.

For large $x$, we have

$$\sqrt{37+70x+x^2} ~\approx~ |x|\tag 1$$

in the sense that the quotient of the functions on either side approach $1$ as $x\to\pm\infty$. $\def\sign{\operatorname{sign}}$

This means $$\begin{align} y &= \frac{20x^2 + 14x + 2}{7m+11xm-m\sqrt{37+70x+x^2}}\\ &\stackrel{(1)}\approx \frac{20x^2 + 14x + 2}{7m+11xm-m|x|}\\ &\stackrel{(2)}= \frac{20x + 14 + 2/x}{7m/x+11m-m\sign x}\\ &\stackrel{(3)}\approx \frac{20x+14}{m\cdot(11-\sign x)} \\ \end{align}$$

provided $m\neq0$. Step (2) divides all terms by $x$ and uses $|x|=x\sign x$. Step (3) neglects all terms of the form $\mathit{const}/x$ because they tend to 0 as $x\to\pm\infty$.

Taking it all together, we have that $$ y\approx \begin{cases} \dfrac{10x+7}{5m}, &\text{ for large positive } x \\ \dfrac{10x+7}{6m}, &\text{ for large negative }x \end{cases}$$ Again, "$\approx$" implies that the quotient of the functions on either side tend to 1. And the even stronger statement holds that their difference tend to 0 as their values tend to $+\infty$ resp. $-\infty$.

emacs drives me nuts
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1

You can proceed in the same way you act when you study if a function has asymptotes. Asympotes are straight lines of the form $r: Y = ax + b$, where

$$a = \lim_{x\to +\infty} \frac{f(x)}{x}$$

$$b = \lim_{x\to +\infty} f(x) - ax$$

Provided that $a\in\mathbb{R}\backslash\{0\}$ and $b\in\mathbb{R}$.

In your case:

$$a = \lim_{x\to +\infty} \frac{\frac{20x^2 + 14x + 2}{7m+11xm-m\sqrt{37+70x+x^2}}}{x} = \frac{2}{m}$$

(I let you to do the math, as an exercise).

$$b = \lim_{x\to +\infty} \frac{20x^2 + 14x + 2}{7m+11xm-m\sqrt{37+70x+x^2}} - \frac{2x}{m} = \frac{7}{m}$$

So you get your asymptote:

$$Y = \frac{2}{m}x +\frac{7}{m}$$

Which shows you, being $m$ a constant, that $y$ behaves linearly.

Enrico M.
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  • Hey sorry, I'm still a little lost. How did you arrive at the value of 2/m and 7/m? For the former, after dividing by x, there is still an x in the numerator – Mui Aug 17 '22 at 02:19
  • @Mui I will add some details later, unless you're already satisfied wih other user's answer! – Enrico M. Aug 17 '22 at 08:56
  • I understand the other user's answer! Thanks for your help :) – Mui Aug 17 '22 at 15:40