Limits as x tends to infinity

Notation

Intuitively we can understand that as x gets larger and larger, 1/x gets smaller and smaller. The limit of 1/x as x tends to infinity is zero. We write this as:

\[\lim_{x\rightarrow \infty}\frac{1}{x}=0\]

Note that an equality sign is used, the limit is equal to zero.

Another way of writing it is:

\[\frac{1}{x}\rightarrow 0 \text{ as } x\rightarrow \infty \]

Here we use arrows instead, 1/x is never equal to zero, but it tends to zero.

Do not mix "lim" and arrows, or expressions and equality-sign; choose one of the forms above!

In the general case we call the limit A, and write it as

\[\lim_{x\rightarrow \infty}f(x)=A \]

The exact definition of a limit is not in the syllabus. Informally it means that the value of \(f(x)\) can be made as close to A as we want, if we just choose x large enough.

Horizontal asymptotes

If a function \(f(x)\) has the limit A as x tends to infinity, then the graph of \(f(x)\) will get closer and closer to the line \(y=A\). The line \(y=A\) is called a horizontal asymptote to \(f(x)\).

Exercises

Graph following functions to find the horizontal asymptote (if there is one) as x→∞

1. \(\displaystyle{ f(x)=\frac{3x^3-x+1}{2x^3+2x^2} }\)
2. \(\displaystyle{ f(x)=\frac{1-2x}{1+2x} }\)
3. \(\displaystyle{ f(x)=\frac{1-2x}{1+2x^2} }\)
4. \(\displaystyle{f(x)=\frac{1+2x^2}{1+2x} }\)
5. \(\displaystyle{f(x)=\frac{x^2+x+1}{x^3+x^2+x+1} }\)
6. \(\displaystyle{f(x)=\frac{4x^4+x^3}{5x^4+x^2+1} }\)
7. \(\displaystyle{f(x)=2+\frac{\sin x}{x} }\)
8. \(\displaystyle{f(x)=2+\frac{x}{\sin x} }\)
9. \(\displaystyle{f(x)=2+\frac{x}{2+\sin x} }\)
10. \(\displaystyle{f(x)=2+\frac{1}{\sqrt{x}-100\cos x} }\)

Is there a way of finding the horizontal asymptote of a rational function (what’s a rational function?) without using an electronic device?

Can you figure out the limits 7-10 without plotting the graphs?