Some limits are really easy to calculate. Some times we can just plug the limit value into the formula and get a result right away. For example, the formula for a horizontal line k units above zero is

$$f(x) = k$$

The limit as x approaches any number is simply k:

$$\lim_{x\rightarrow anything} f(x) = k$$

The limit of a constant function k at any point is k.

Slightly more complicated, lets take the formula for a straight line:

$$f(x) = 3x + 4$$

If we want the limit as x approaches 14, we just plug the number 14 into the formula, and do the arithmetic:

$$\lim_{x\rightarrow 14} f(x) = 3(14) + 4 = 46$$

Sometimes, like when we were calculating the slope of a tangent,

$${f(x_0 + d) - f(x_0)} \over { d }$$

we can't just plug in the number because we would be dividing by zero:

$${\lim_{ d \rightarrow 0 }} {{ f( x_0 + d ) - f( x_0 ) } \over d } $$

The simplest example of this is the formula

$$f(x) = {1 \over x}$$

We can see from the graph:

that

$${\lim_{ x \rightarrow \infty }} {1 \over x } = 0$$

but it also looks like

$${\lim_{ x \rightarrow 0 }} {1 \over x }  = \infty$$

However, infinity is not a limit (sort of by definition, don't you think?) so we say

$$ {\lim_{ x \rightarrow 0 }} {1 \over x }  $$

does not exist.

A particularly fun limit with the denominator going to zero is:

$${\lim_{ x \rightarrow 0 }} {sin(x) \over x }  = 1$$