Asymptote Formula

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity. There are two types of asymptote: one is horizontal and other is vertical. Below mentioned is asymptote formula.

Asymptote Formula

Solved Examples

Question 1: Find the asymptotes for $f(x) = \frac{2x^{2} + 2x}{x^{2}+1}$

We can see at once that there are no vertical asymptotes as the denominator can never be zero.
$x_{2}$ + 1 = 0
$x_{2}$ = –1 has no real solution
Now see what happens as x gets infinitely large:\[\lim_{x\rightarrow\infty}\;\frac{2x^{2}+2x}{x^{2}+1}\]

The method we have used before to solve this type of problem is to divide through by the highest power of x.

= $\LARGE\lim_{x\rightarrow\infty}\;\frac{2x^{2}+2x}{x^{2}+1}$

= $\LARGE \frac{\frac{2x^{2}}{x^{2}} + \frac{2x}{x^{2}}}{\frac{x^{2}}{x^{2}} + \frac{1}{x^{2}}}$

= $\LARGE\frac{2+0}{1+0}$

= $\LARGE2$

More topics in Asymptote Formula
Slant Asymptote Formula

Practise This Question

A bowling ball is thrown down the alley in such a way that it slides with a speed v0 initially without rolling. What will it it's speed be when (and if) it starts pure rolling? The transition from pure sliding to pure rolling is gradual, so that both sliding and rolling takes place during this interval.