Generally when we take the limit of a function $f(x)$ as $x \to \infty$, we get a constant value or $\pm \infty$. However, for some functions, as $x$ grows large, the curve approaches that of another function. For example, for large positive and negative $x$, the curve $y=x^2+\frac{1}{x}$ approaches $y=x^2$. Is this often used in the study of limits? I can see how it could be useful when a function approaches $\infty$ because it gives one a simple idea of how quickly it approaches infinity.
Thanks!
https://en.wikipedia.org/wiki/Asymptotic_analysis
This is what we call equivalents, and the notation $f(x)\sim g(x)$ means (provided $g(x)\neq 0$) that $\lim\limits_{x\to+\infty} \frac{f(x)}{g(x)}=1$ or in the $o$ notation $f(x)=g(x)+o(g(x))$.
From this definition it is obvious that if $g$ has a limit (including values $\pm\infty$), then $f$ has one too, and it is the same.
Of course it is not useful only for equivalents at infinity, you can have equivalents in $0$ or at any desired point.
In your case we would say $f(x)=x^2+\frac 1x\sim x^2$ to say that at infinity $f(x)$ behaves or is not much different from $x^2$.
In $o$ notation this means $f(x)=x^2+o(x^2)$.
Obviously you can multiply and divide equivalents, but not add them since negligible terms tend to vanish.
For instance at infinity $\displaystyle f(x)=\frac{3x^2+-5x+7}{7x^3-\ln(x)}\sim\frac{3x^2}{7x^3}\sim \frac{3}{7x}$ we have taken equivalents for numerator and denominator and divided them without restriction.
And at $0^+$ we have $f(x)\sim\frac{7}{-\ln(x)}$.
In any case with equivalents you keep only the main term, negligible ones are simply cancelled.
So this is a tool for finding limits and asymptotics quickly.
Excepted that, equivalents are compliant with integrals and series of terms constant sign.
For instance if $a_n>0$ and $a_n\sim b_n$ and $\sum b_n$ converges then $\sum a_n$ converges too.