This limit is of the indeterminate form . Combining the two dfractions in this limit should lead to a cancellation of the factor giving rise to this indeterminacy, i.e.
Cancelling out from the numerator and the denominator.
Before trying to solve this, try to feel that this expression will have a finite limit even though the number of factors being multiplied tends to infinity. This is because the successive factors become closer and closer to and their ‘contribution’ to the final product becomes smaller and smaller Now, to simplify this product, we multiply it by . This is what happens:
Hence, the value of the limit is
The numerator and denominator both tend to as because of the common factor .
Hence, factorization leads to :
Cancelling out from both the numerator and the denominator
There is still another common left in both the numerator and the denominator.
Factorization again leads to
Cancelling out again from the numerator and the denominator.
Evaluate the following limits:
This limit can evidently be solved by rationalising both the numerator and the denominator.
This can be solved by rationalisation again.
This limit is of the indeterminate form (and look very complicated !)
However, division of both the numerator and denominator by directly reduces the limit to a determinate form.
and these two terms tend to as
since the denominator is , we can get a hint that the numerator