If we are trying to use limit laws to compute this limit, we would first have to use the Product Law to say that We are only
allowed to use this law if both limits exist, so we must check this first. We know from continuity that However, we also know that
is badly behaved as approaches ; going to infinity on one side and zero on the other. Thus the limit does not exist. Therefore, we
cannot use the Product Law.
Can this limit be directly computed by limit laws?
Can this limit be directly computed by limit laws?
If we are trying to use limit laws to compute this limit, we would like to use the Quotient Law to say that We are only allowed to
use this law if both limits exist and the denominator is not . We suspect that the limit in the denominator might equal , so we
check this limit.
This means that the denominator is zero and hence we cannot use the Quotient Law.
Can this limit be directly computed by limit laws?
Compute:
If we are trying to use Limit Laws to compute this limit, we would now have to use the Quotient Law to say that We
are only allowed to use this law if both limits exist and the denominator is not . Let’s check the denominator and numerator
separately. First we’ll compute the limit of the denominator:
Therefore, the limit in the denominator exists and does not equal . We can use the Quotient Law, so we will compute the limit of the numerator: Hence
Can this limit be directly computed by limit laws?
If we are trying to use limit laws to compute this limit, we would have to use the Product Law to say that We are only allowed to use this law
if both limits exist. Let’s check each limit separately.
If we are trying to use limit laws to compute this limit, we would now want to use the Composition Law to say that But we are only allowed to use this law if is continuous at . So we need to compute this limit to determine if is continuous at that value. We can do this using the Product Law.
So, since is , and doesn’t even exist (let alone be continuous) at , we cannot apply the Composition Law.
Can this limit be directly computed by limit laws?
If we are trying to use limit laws to compute this limit, we would have to use the Product Law to say that We are only allowed to
use this law if both limits exist. We know , but what about ? We do not know how to find using limit laws because is not in the
domain of .
Can this limit be directly computed by limit laws?
Compute:
If we are trying to use limit laws to compute this limit, we would have to use the Quotient Law to say that We are
only allowed to use this law if both limits exist and the denominator does not equal . Let’s check each limit separately, starting
with the denominator
On the other hand the limit in the numerator is
The limits in both the numerator and denominator exist and the limit in the denominator does not equal , so we can use the Quotient Law. We find: