What is the difference between a function and its limit?

In calculus, a function and its limit are related but distinct concepts. Let's explore the difference between the two:

• Function: A function, denoted as $f(x)$, is a rule that assigns to each input $x$ in its domain exactly one output $f(x)$. For example, $f(x) = x^2$ is a function that takes any real number $x$ and returns $x$ squared. Functions are defined for all values in their domain and describe the behavior of the output as $x$ changes.

• Limit: A limit, on the other hand, is a concept that describes the behavior of a function as its input approaches a specific value. It's denoted as $\lim_{x \to a} f(x) = L$. This means that as $x$ gets closer and closer to $a$, the value of $f(x)$ gets closer and closer to $L$, but $f(x)$ does not necessarily have to equal $L$ when $x = a$. In other words, the limit describes what happens to the function's output as its input approaches a certain value, not what the function's output is at that exact value.

Here's a simple example to illustrate the difference:

Consider the function $f(x) = \frac{1}{x}$. This function is defined for all real numbers except $x = 0$. Now, let's find the limit as $x$ approaches 0:

$\lim_{x \to 0} \frac{1}{x} = \lim_{x \to 0} \frac{1}{x} \cdot \frac{-x}{-x} = \lim_{x \to 0} \frac{-1}{x^2} = -\infty$

So, the limit of $f(x)$ as $x$ approaches 0 is $-\infty$, which means that as $x$ gets closer and closer to 0, the value of $f(x)$ gets closer and closer to $-\infty$. However, the function $f(x) = \frac{1}{x}$ is undefined at $x = 0$, and it never actually reaches $-\infty$. This is the key difference between a function and its limit: the limit describes the behavior of the function as $x$ approaches a value, while the function describes the output for all values in its domain.