I'm trying to intuitively grasp the following definition:
The real-valued functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$ if $$\lim_{x \to \infty} \dfrac{f(x)}{g(x)}=1.$$ We write this as $f \sim g$.
My question is: how do we visually interpret this in terms of the graphs of $f$ and $g$? Does this mean that the graphs of $f$ and $g$ get closer to each other as $x$ gets larger and larger?
My only intuition for this comes from the following example: we know that $\sin x \sim x$ as $x \to 0$ (since $\lim_{x \to 0} (\sin x)/x = 1$). And as we can see below, the graphs of $\sin x$ (the green line) and $x$ (the black line) get closer and closer as $x$ goes to $0$.
But this intuition does not seem to hold for functions asymptotically equivalent at $\infty$. I graphed $x^2 + x$ (black line) and $x^2$ (green line) and their graphs do not appear to be getting closer at all! In fact, it looks like there's a "gap" between the two graphs.
This leads me to believe that I'm not interpreting "asymptotically equivalent" in the right way. I've come across the idea that $f \sim g$ means that $f$ and $g$ have the "same rate of growth", but that feels very unintuitive for me. Is there are a way to see that in the graphs?
Any guidance would be greatly appreciated! Thanks.
Best Answer
$x^2+x$ and $x^2$ are asymptotically equivalent, since $$ \lim_{x \to \infty}\frac{x^2+x}{x^2}=\lim_{x \to \infty}1+\frac{1}{x}=1 \, . $$ So it is probable that you simply didn't choose $x$ values that were large enough to make your intuitions work. For example, $x=100$ gives $$ \frac{100^2+100}{100^2}=\frac{10100}{10000}=1.01 \, , $$ and you can see that they are very close to each other in relative terms. Notice the use of the word relative here. The example you gave helps illustrate what I mean. Let $f(x)=x^2+x$ and $g(x)=x^2$. $f(x)-g(x) \to \infty$ as $x \to \infty$. This means that in absolute terms, the two functions are growing apart. However, the ratio between them—what you need to multiply $x^2$ by to get $x^2+x$—is approaching $1$. This is what the notion of 'asymptotically equivalent' is trying to capture.
To visualise this, it would be better to use a logarithmic scale:
Look at $x=10$, for instance. $x^2=100$, whereas $x^2+x=110$. This discrepancy looks small when we use a logarithmic y-axis, since the gap of $10$ is small relative to how large the functions are. Or, as Paramanand Singh has suggested, we could plot $y=(x^2+x)/x^2$: