Consider the following limit.
lim x→2 
4x + 3 
. 
If you estimate the limit either numerically or graphically, you will find that
lim x→2 
4x + 3  ≈ −0.1818 
But, notice that you can obtain this answer by simply substituting x = 2 in the given function:
f(x) = 
4x + 3  
f(2) = 
8 + 3  =    11  =  0.181818... 
This answer is more accurate than the one coming from numerical or graphical method; in fact, it gives the exact limit.
Q Is that all there is to evaluating limits algebraically: just substitute the number x is approaching in the given expression?
A Not always, but this often does happen, and when it does, the function is continuous at the value of x in question. Recall the definition of continuity from the previous tutorial:
Continuous Functions
The function f(x) is continuous at x = a if
The function f is said to be continuous on its domain if it is continuous at each point in its domain. If f is not continuous at a particular point a, we say that f is discontinuous at a or that f has a discontinuity at a. 
Q How do you tell if a function is continuous?
A As we saw in the previous tutorial, we can tell whether a function is continuous by looking at its graph. If the graph breaks at some point in the domain, then f has a discontinuity there. If the function is specified algrabcially, sometimes it is easy to tell whether it is continuous by just looking at the formula:
A closedform function is any function that can be obtained by combining constants, powers of x, exponential functions, radicals, absolute values, trigonometric functions, logarithms (and some other functions you may never see) into a single mathematical formula by means of the usual arithmetic operations and composition of functions. Examples of closedform functions are:





They can be as complicated as you like. The following is not a closed form function.
f(x) =  1 if x < 1  
x^{2} + x if 1 ≤ x ≤ 1  
2  x if 1 < x ≤ 2 
The reason for this is that f(x) is not specified by a single mathematical expression. What is nice about closedform functions is the following.
Continuity of Closed Form Functions
Every closed form function is continuous on its domain. Thus, the limit of a closedform function at a point in its domain can be obtained by substitution. Example
is a closed form function, and x = 2 is in its domain. Therefore we can obtain lim_{x→2} f(x) by substitution:
as we saw above. 
Q What if f(x) is a closed form function, but the point x = a is not in the domain of the function?
A Then, you either:
Let us evaluate
lim x→2 
x + 2 
Ask yourself the following questions:
2. Is the value x = a in the domain of f(x)?
Therefore, we consult the above Question/Answer discussion, and simplify the function, if we can.
x + 2  = 
 
=  3x5. 
Since we are now left with a closed form function that is defined when x = 2, we can now evaluate the limit by substitution:
lim x→2 
x + 2 
=  lim x→2 
3x  5  =  3(2)5 = 11. 
Back in the tutorial on functions from the graphical point of view, we looked at the following function:
f(x)  = 

This time, we are not showing you the graph right away, and ask you to look at the formula instead. Notice:
Now try the rest of the exercises in Section 3.3 in Applied Calculus or Section 10.3 in Finite Mathematics and Applied Calculus