Notice that as , , that is , has no particular limit to which it converges. Hence keeps oscillating between and as becomes smaller and smaller, i.e., Therefore, the limit for this function does not exist.
This is also clear from the graph (approximate) of sketched below:
In this limit, in addition to , ‘’ is also present. Thus, although remains oscillating and does not approach any particular limit, it nevertheless remains somewhere between and , and when it gets multiplied by (where ), the whole product gets infinitesimally small.
Again, this is evident from the graph below:
This limit can be evaluated purely by observation as follow:
Although and are both tending to infinity, increases very slowly as compared to .
For example, when , is just . When (a very large number indeed !), is just .
Therefore, decreases and becomes infinitesimally small as , i.e.,
(We can also use the rule to evaluate the limit above: this rule will be discussed later)
As , , so that this limit is of the indeterminate form
But as in parts () and (), try to see that the product becomes infinitesimally small as .
For example, at , and =
At , (which is very very small)
Hence, here again,
If then as , and , so that the limit is .
For also, the limit is obviously .
For we write as
Now, since is finite, let be the integer just less than or equal to
The product of the first terms is finite; let it be equal to .
The product inside the limit consists of all terms less than . Also successive terms become smaller and smaller and tend to as .
Therefore, this product tends to and hence the value of the overall limit is
Note: As we mentioned earlier, once we have studied differentiation, we’ll study the L’Hospital’s rule for evaluation of limits of the form . However, it might be useful to know the rule right away – so we provide a brief idea here:
As if and both tend to or both tend to infinity, then if the latter limit exists
Here are two examples:
This rule is simple yet extremely powerful, and in general, you’ll be able to solve most limits using this rule.