Graphic intersecting asymptotes

Question

Sometimes graphics intersect the asymptotes(horizontal) of the function we plot and then they tend to the asymptote to infinity.What gives us the information whether the graph only tends to the asymptote and does not intersect it or intersects the asymptote a then tends to her for a given point?

Answer 1

Those two issues are quite separate: (1) whether a function's graph intersects a horizontal line, and (2) whether the horizontal line is an asymptote to the function's graph.

Let us say that the function is $y=f(x)$ and the horizontal line is $y=b$. You find if they intersect by solving the equation $f(x)=b$. You find if the line is an asymptote by checking if either $\lim_{x \to -\infty}f(x)=b$ or $\lim_{x \to +\infty}f(x)=b$.

Some examples: The function $f(x)=\frac 1x$ has the horizontal asymptote $y=0$ but does not intersect it. It does intersect $y=1$, which is not an asymptote.

The function $f(x)=\frac{\sin x}x$ also has $y=0$ as a horizontal asymptote, but here the function does intersect the line (infinitely many times). It intersects $y=\frac 12$ a few times but does not intersect $y=2$ at all.