# Asymptotes in Graphs

## Asymptotes

A straight line is called an asymptote to the curve *y* = *f *(*x*) if, in layman’s term, the curve touches the line at infinity (this is not technically correct; we should say that the curve *tends to* touch the line as infinity is approached or as \(x \to \infty \))

More accurately, an asymptote to a curve is a line such that the distance from a variable point M on the curve to the straight line approaches zero as the point M recedes to infinity along some branch of the curve.

Referring to the previous example, we see that *y* = 0 is a horizontal asymptote to \(\begin{align}f\left( x \right) = \frac{x}{{1 + {x^2}}};\end{align}\) *x* = 1 and *x* = 2 are vertical asymptotes to \(\begin{align}f\left( x \right) = \frac{1}{{{x^2} - 3x + 2}}\end{align}\) and *y* = 1 is a horizontal asymptote to \(\begin{align}f\left( x \right) = \frac{{{x^2} - x + 1}}{{{x^2} + x + 1}}\end{align}\). Of course, there can be inclined asymptotes also. We now formally distinguish between the three kinds of asymptotes and outline the approach to determine them.

**(a) Horizontal asymptotes**: If \(\mathop {\lim }\limits_{x \to \pm \infty } f\left( x \right) = k\) then *y* = *k* is a horizontal asymptote to *f *(*x*).

**(b) Vertical asymptotes**: If LHL or RHL (or both) at *x* = *a* are infinity for *f *(*x*), then

*x* = *a* is a vertical asymptote to *f *(*x*).

**(c) Inclined asymptotes**:** **If and \(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}{x} = {a_1}\end{align}\), then \(\begin{align}y={{a}_{1}}x+~{{b}_{1~}}\end{align}\) is an inclined right asymptote to *f *(*x*).

Similarly, if \(\begin{align}\mathop {\lim }\limits_{x \to - \infty } \frac{{f\left( x \right)}}{x} = {a_2}\end{align}\) and \(\mathop {\lim }\limits_{x \to \infty } \left( {f\left( x \right) - {a_1}x} \right) = {b_1}\), then \(y = {a_2}x + {b_2}\) is an inclined left asymptote to *f*(*x*).

This discussion will become more clear with an example.

Let \(\begin{align}f\left( x \right) = x + \frac{1}{x}\end{align}\)

Now, \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = \infty \,\,\,{\rm{and}}\,\,\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = - \infty \)

\( \Rightarrow \,\,\, x = 0\) is a vertical asymptote to \(f\left( x \right)\).

\(\begin{align}\mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}{x} = 1; \mathop {\lim }\limits_{x \to - \infty } \frac{{f\left( x \right)}}{x} = 1\end{align}\)

\(\mathop {\lim }\limits_{x \to \pm \infty } \left( {f\left( x \right) - x} \right) = 0.\,\,\,\left( {\because a = 1,\,b = 0} \right)\)

\( \Rightarrow \quad y = x\) is an inclined asymptote to *f*(*x*).

The graph is sketched below. The extremum points are \(x = \pm 1:\)

Before closing this section with a few more examples, have are a few general steps* to be followed whenever one is encountered with the task of sketching the graph of an arbitrary function *f *(*x*):

**(i) ** Find the domain of the given function.

**(ii)** Determine more of its characteristics; for example, is the function even or odd or neither? Is it periodic? If yes, what is the period? And so on.

**(iii)** Test the function for continuity and differentiability.

**(iv)** Find the asymptotes of the graph, if any.

**(v)** Find the extremum/ inflexion points and the intervals of monotonicity.

**(vi) ** To improve accuracy of the plot, one can always evaluate *f *(*x*) at additional points.

This is a very general sequence and mostly the graph would be able to be plotted without necessarily following all the steps. For all our current purposes, these steps are more than sufficient.

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