How to Interpret and Calculate Limits of a Function

Limits are values that say something about what happens to an expression when a variable approaches a certain value. The value of the limit can be $-\infty$, $\infty$, or any number on the number line.

We use a specific notation for limits:

$$\underset{x\to a}{\lim }\mathit{\text{expression}}$$

You read this as “the limit of expression as $x$ approaches $a$”, or “the limit of expression as $x$ tends to $a$”.

Here, “$\lim$” is an abbreviation for “limit”. Mathematicians try to make things as simple and intuitive as possible.

Let $f(x)$ be a polynomial of degree $n$. Thus

If $x$ approaches plus or minus infinity, then $a_n{x}^n$ eventually becomes the dominant term in the expression, and it will determine whether the function value of $f(x)$ is positive or negative.

Consider the limit of $f(x)$ when $x\to \infty$. The dominant term affects the sign of the function value of $f(x)$, since $x^n$ tends to positive infinity if $n$ is an even number, and it tends to negative infinity if $n$ is an odd number. Below is a summary of the different cases:

Note! When you have a simple fraction $\frac{1}{x}$ and $x\to \infty$, then $\frac{1}{x}=0$. All fractions where $x$ is only found in the denominator approach 0 when $x$ approaches infinity. Neat!

Note! $\infty$ is not a number! So no matter how big a number you choose on the number line, $\infty$ is infinitely larger. This means that the ratio between $x$ and $a$ when $x\to \infty$ or $x\to -\infty$ is always infinitely large.