Limits describe the behavior of functions at certain points or regions.

For our syllabus, we will mainly employ limits as a tool to explore “infinity behavior”.

Rational functions

Let us consider the rational function $\displaystyle y = \frac{1}{x}.$

When $x$ gets very large, $\displaystyle \frac{1}{x}$ gets exceedingly small and arbitrarily close to $0.$ We thus say that $\displaystyle \frac{1}{x} \to 0$ when $x \to \infty.$

In limit notation, ${\displaystyle \lim_{x \to \infty} \frac{1}{x} = 0.}$

On the other hand, while we cannot substitute $x=0$ into the function, we can explore the behavior of it when $x$ gets exceedingly small and close to zero. $y$ will then get exceedingly large without bounds. We can thus say that $\displaystyle \frac{1}{x} \to \infty$ when $x \to 0^+$ where the plus symbol indicate that we are going to $0$ “from the positive side”. (When $x \to 0^-$, then $y \to -\infty.$)

In limit notation, ${\displaystyle \lim_{x \to 0^+} \frac{1}{x} = \infty.}$

Proper fractions

Let $f(x)$ and $g(x)$ be polynomials and consider the rational function ${\displaystyle y=\frac{f(x)}{g(x)}}$.

We say that ${\displaystyle \frac{f(x)}{g(x)}}$ is proper if the largest degree of $f$ is strictly smaller than $g$.

For example, ${\displaystyle \frac{5}{2x+3}}$ and ${\displaystyle \frac{2x+5}{x^2-4x+3}}$ are proper while ${\displaystyle \frac{x^2-4x+3}{2x+5}}$ and ${\displaystyle \frac{x-1}{2x+5}}$ are improper.

If the rational function ${\displaystyle y=\frac{f(x)}{g(x)}}$ is proper, then ${\displaystyle \frac{f(x)}{g(x)} \to 0}$ as $x \to \infty$.

The limits when $x \to \infty$ for improper fractions can be handled by long division.

Other common limits

• As ${x \to -\infty,}$ ${\mathrm{e}^x \to 0.}$
• As ${x \to 0^+,}$ ${\ln x \to -\infty.}$
• As ${x \to \frac{\pi}{2}^-,}$ ${\tan x \to \infty.}$