The concept of a limit in mathematics, particularly in calculus, describes the value that a function or sequence approaches as the input approaches some value.

Limit of a Function:

For a function $f(x)$, the limit is written as:

$\lim_{{x \to a}} f(x)$

This reads as "the limit of $f(x)$ as $x$ approaches $a$." It represents the value that $f(x)$ gets closer to as $x$ gets closer to $a$, without necessarily reaching $a$.

Example:

$\lim_{{x \to 2}} (3x + 1)$

To solve: $\lim_{{x \to 2}} (3x + 1) = 3(2) + 1 = 7$

Limit at Infinity:

Sometimes, we are interested in the behavior of a function as $x$ approaches infinity. For example:

$\lim_{{x \to \infty}} \frac{1}{x} = 0$

As $x$ gets larger and larger, $\frac{1}{x}$ gets closer to 0.

Types of Limits:

1. Finite Limit: The function approaches a specific finite value as $x$ approaches some value.
2. Infinite Limit: The