### Limit: Definition

The limit of a function f is the value the function approaches when its argument approaches a certain value.

Mathematically, we write

\lim \limits_{x \to a} f(x) = l

We say that f tends towards l when x tends towards a.

Depending on the case, a and l may be real numbers, or equal to \pm \infty.

### Limits of standard functions

#### - Real-valued finite limits

For example, for f(x) = x^2, the limit when x tends towards 2 is 2^2 = 4.

#### - Infinite limits

When the function is well defined at a real number a (it is said to be continuous at a), then the limit at a is exactly f(a).

When the variable x takes on very large values (positively or negatively), we say that x tends towards more or less infinity. In this case, we distinguish between cases where f(x) approaches a finite value and those where f(x) moves away towards infinity. In the former case, f is said to converge, and in the latter, f to diverge.

Below is a table summarizing the limits of common functions at infinity.

When a function admits a finite limit l \in \mathbb{R} when x \to \infty, then we say that f admits a hor

#### - Infinite limits at a finite value

\begin{array}{|c|c|}
\hline \text{Function} & \text{Limit} 0, \: x \to 0 \\
\hline f(x) = \frac{1}{x} & \begin{cases} & +\infty \text{ si } x \gt 0 \\ & \infty \text{ si } x \lt 0 \end{cases} \\
\hline nan & nan \\
\hline f(x) = \ln(x) & -\infty \\
\hline f(x) = \sqrt{x} & 0 \\
\hline
\end{array}

When \lim_{x \to a} f(x) = \pm \infty, with a a real number, then the curve of f is said to have a vertical asymptote at a. In other words, the curve moves closer and closer to the straight line with equation x=a.

### Determining a limit in practice

To determine limits, it's sometimes simpler to use other functions.

Comparing functions allows you to compare their limits, when they exist.

Comparing functions allows you to compare their limits, when they exist.

There are two main properties for comparing functions

#### - Comparison theorem

Let f and g be two functions defined on an interval I = [a, +\infty[.

Assume that for any x \in I ; f(x) \geq g(x)

Assume that for any x \in I ; f(x) \geq g(x)

If \lim_{x \to +\infty}g(x) = +\infty, so \lim_{x \to +\infty}f(x) = +\infty

If \lim_{x \to +\infty}f(x) = -\infty, so \lim_{x \to +\infty}g(x) = -\infty

#### - Gendarme theorem

Let f, g and h be three functions
Assume that for any x \in I ; g(x) \leq f(x) \leq h(x)

If g and h tend to the same limit l in +\infty, then \lim_{x \to +\infty}g(x) = +\infty

You now know what a limit is and how to calculate it. Calculating limits allows you, for example, to define the derivative of a function as the limit of a rate of increase.

If you'd like to discover all the mathematical concepts involved in data science, we invite you to take a look at our curricula.

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