what is difference between study of asymtotic and limit?

Question

In asymptotic, what properties of function we study and in limits what properties of function we study?

Do both describe limiting value of function(s), if yes then what information or knowledge they suppress (or hide or consider unwanted)?

Answer 1

The limit gives you a single value. The asymptotics gives you a good approximation that is a function of the independent variable.

For example

$$\sqrt{x+2}-\sqrt{x}$$ tends to zero, but by

$$\sqrt{x+2}-\sqrt{x}=\frac2{\sqrt{x+2}+\sqrt{x}}\sim\frac1{\sqrt x},$$ we know how "fast".

Answer 2

A limit is a value. It representants the number you will approach when the parameter of the function you study will grow to a certain level. For example $$ \frac{1}{x}+4 \underset{ x \rightarrow +\infty}{\rightarrow}4 $$ Then the graph of the function is near $4$ as $x$ grows and tends be to be equal to it.

In asymptotic expression, you study how the function behaves around a certain value. And usually compare to another function you know, for example $$ f(x)=\frac{x^2+3}{x+1} $$ Observe that it has an asymptot $g :x \mapsto x-1$, it represents the two graphs. As $x$ grows, they behave the same !

Note that an asymptotic expression is more precise than just a limit, for example $$ \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \underset{+\infty}{\sim}1-2e^{-2x} $$ hence we conclude that $$ \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\underset{ x \rightarrow +\infty}{\rightarrow}1 $$ But the graph will be as near $1$ as $x \mapsto e^{-2x}$ is near to $0$ when $x$ grows.