Both the limits above are indeterminate, of the form . We are discussing here a geometric interpretation of these limits. Consider a sector of a unit circle as shown in the figure below.
We see that
What happens as decreases or as
We see that the difference between the three areas considered above tends to decrease;
(but remains less than )
(remains greater than )
Consider the limits below and you should understand:It is important to observe how any function approaches a limit. For example, in the case above, as approaches from the left side while approaches from the right side. This makes a big difference. Why
Consider the expression for . As gets larger and larger or as , the base gets closer to while the exponent (), tends to infinity. Hence, this limit is of the indeterminate form . Its very important to get a ‘feel’ that the value will converge to a fixed, definite value, as increases. You should get this feel by looking at the table below.
Try to show that the limit of is bounded and lies between and and , that is, (you can use the binomial theorem in conjunction with the sandwich theorem to prove this, by first proving it for an integral )We see that as becomes larger, the term converges to some value (This can be proved) This limiting value is denoted by . is an irrational number and its value is .
The limit we have just seen is extremely important and will be widely used subsequently.