In order to understand working of this rule, let us consider a radical :
We are required to divide the radical by x, when it is known to be negative. Following the fact stated above,
Problem : Determine limit :
Solution : Here, indeterminate form is ∞/∞. Dividing each term by
This is determinate form. As
and as
. Hence, limit is :
Alternatively, we can employ second method to evaluate limit. Taking out
Problem : Determine limit :
Solution : Here, indeterminate form is ∞/∞. By inspection, we see that a/b>1 and b/a<1. As we know x→∞,
→ 0 if c<1. Hence, we are required to get terms in the form b/a raised to some power. Dividing numerator and denominator by
, we have :
This is determinate form. As
. Hence,
Problem : Determine limit :
Solution : The indeterminate form is ∞/∞. Writing expression of sum of square of natural numbers, we have :
As
Problem : Determine limit :
Solution : Here, indeterminate form is ∞-∞. Rationalizing, we have :
Dividing by x. Note x is negative. Hence
As
.
Exercises
Determine limit
Here, indeterminate form is ∞ - ∞. We simplify to change the form of expression from determinate ,
This form is determinate. Plugging “1” for “x”, we have :
Determine limit
Here, indeterminate form is ∞-∞. Rationalizing, we have :
Dividing by
,
As
Determine limit
Here, indeterminate form is ∞-∞. Rationalizing surd, we have :
Dividing each of terms by x, we have :
This is determinate form. As x->∞, 1/x ->0.
Determine limit :
Here, indeterminate form is ∞-∞. Rationalizing surds, we have :
Dividing numerator and denominator by
,
This is determinate form. As
,
Determine limit
Here, indeterminate form is 0/0. We put
.
Using formulae :
Determine limit
Here, indeterminate form is ∞ - ∞. We simplify to change indeterminate form and find limit,
It is determinate form. Plugging “1” for “x”, we have :
Determine limit :
Here, indeterminate form is ∞/∞. We divide each term by
.
As
, and
Hence,
Determine limit :
Here, indeterminate form is 0/0. Using standard form,
This is determinate form.
Determine limit :
Here, indeterminate form is 0/0. We simplify to change indeterminate form and find limit,
This is determinate form. Plugging “2” for x, we have :
Determine limit :
Here, indeterminate form is 0/0. Rationalizing surds,
Each term limits to
.
This is determinate form.
Determine limit
Here, indeterminate form is 0/0. Using standard formulae, we have :
It is determinate form. Evaluating, we have :
Determine limit :
Here, indeterminate form is ∞-∞. Rationalizing surds,
This is in determinate form. Plugging “1” for “x”, we have :
Determine limit :
Here, indeterminate form is 0/0. The numerator and denominator tend to 0 as x->1. It means (x-1) is factor of both numerator and denominator. Dividing polynomials (long method or otherwise) and using quotient :
This is determinate form. Plugging “1” by “x”, we know :
Determine limit :
Indeterminate form is 0/0. We simplify the expression to change indeterminate form and find limit,
This is not in indeterminate form. Plugging “a” for “x”