Limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

Examples:

• \(\lim_{x \to 2} x = 2\)
• \(\lim_{x \to 5} (x-1) = 4\)
• \(\lim_{x \to 0} x^2 = 0\)
• \(\lim_{x \to 0} \sin(x) = 0\)

Limit of a fraction of two functions equals to the fraction of limits of those two functions.

\begin{equation*} \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)} \end{equation*}

Examples:

• \(\lim_{x \to 3} \frac{x-1}{x+2} = \frac{\lim_{x \to 3}(x-1)}{\lim_{x \to 3}(x+2)} = \frac{2}{5}\)

What if our limit of functions is undefined? In essence, what if our final fraction is \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)?

Examples:

• \(\lim_{x \to 0} \frac{\sin(x)}{x} = \frac{\lim_{x \to 0} \sin(x)}{\lim_{x \to 0} x} = \frac{0}{0}\)
• \(\lim_{x \to \infty} \frac{x}{e^x} = \frac{\lim_{x \to \infty} x}{\lim_{x \to \infty} e^x} = \frac{\infty}{\infty}\)

Let \(f\) and \(g\) be both differentiable (derivatives \(f'\) and \(g'\) exist and are defined), if \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\) or \(\pm \infty\) for all \(x\) in the domain with \(x \neq a\), then

\begin{equation*} \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \end{equation*}

Recall some examples from Motivation

\begin{align*} \lim_{x \to 0} \frac{\sin(x)}{x} & = \lim_{x \to 0} \frac{(\sin(x))'}{(x)'} \\ & = \lim_{x \to 0} \frac{\cos(x)}{1} \\ & = \lim_{x \to 0} \cos(x) = 1 \end{align*}

This means that as \(x\) gets infinitely small, \(\sin(x)\) and \(x\) tend to have the same value!

Our second example from motivation

\begin{align*} \lim_{x \to \infty} \frac{x}{e^x} & = \lim_{x \to \infty} \frac{(x)'}{(e^x)'} \\ & = \lim_{x \to \infty} \frac{1}{e^x} \\ & = \frac{1}{\lim_{x \to \infty} e^x} = 0 \end{align*}

So as \(x\) gets infinitely large, \(e^x\) is getting larger much quicker than \(x\) does!