A limit is defined as the value a function approaches as the variable within that function gets nearer and nearer to a particular value. Suppose we have a limit described as . This indicates the value of when is infinitely close to but not exactly equal to . The substitution rule is a method of finding limits, by simply substituting with . The mathematical manifestation of this rule would be
Let's try out a few examples first.
Find the value of .
This is simple. Just use the substitution rule and plug in , and we have
However, the substitution rule does not always hold. In order to use the substitution rule, the function must satisfy the following condition:
This means that the graph of does not break up anywhere within its domain. An example of a discontinuous function is . Try drawing this. You will notice that the graph breaks up at , and thus it is discontinuous at , so we cannot use the substitution rule when finding . In fact, this limit does not exist at all, but we will discuss this later on.
Find the value of
The figure depicts the graph of the function Observe that the graph is discontinuous at , which means that we cannot apply the substitution rule to find the given limit.
So, now our discussion comes to a simple and explicit conclusion: "If the function is continuous, just substitute the variable with the value it converges to!"
Rewriting the expression, we have
Since as the answer is
Since the denominator approaches infinity, i.e. as the answer is