Limits: The Basic Idea

The intuitive meaning of $\lim\limits_{x \to c} f(x)$

For example: function

$$f(x)= \begin{cases} x-1& \text{if x!=2} \\ 3& \text{if x=2} \end{cases}$$

The graph is:

From the graph, in this function, $f(2)=2$. However, $\lim\limits_{x \to 2} f(x)$ is not 2. It is the only value where x is very very close to 2, not actually 2. From the graph, it is easy to claim that the value where x is close to 2 is 1 ($f(1.999)=0.999, f(1.99999)=0.99999, f(2.001)=1.001, f(2.00001)=1.00001$). So we can write this statement as $$\lim\limits_{x \to 2} f(x) = 2$$ And this statement also could write as $$f(x)\rightarrow 1 \quad \textrm{as} \quad x\rightarrow 2.$$ Also, here is a important things that when you write something like $\lim\limits_{x \to 2} f(x) = 2$, $\lim\limits_{x \to 2} f(x)$ is not a function of x. We could replace x into whatever you want, like $\lim\limits_{q \to 2} f(q) = 2$. The letter “x” is only a temporary label for some quantity that is (in this case) getting very close to 2.

Left-Hand and Right-Hand Limits

If there is a funtion f(x), and the graph is: