Find the limits, Limits of polynomial sequences, Limits of exponential sequences, Limits of sequences with radical.
Let's see what should be done with limits of this type:
Here is a case like that:
The trick is to divide by .
Both the numerator and the denominator.
This way we turned into .
The latter is easier to figure out, we know it is 2.
Let's see another one.
Let's divide this one by , too.
Let's see what we get.
The numerator tends to 4.
The denominator tends to zero.
Well, this is a problem.
The cause of the problem is that the highest degree member in the denominator is quadratic.
So we should not be surprised that if we divide by , then everything in the denominator will tend to zero.
If we don't want the denominator to tend to zero, then we have to divide by the highest degree member of the denominator.
Divide both the numerator and the denominator by the highest degree member of the denominator.
Divide both the numerator and the denominator by the member of the denominator that has the highest base.
First we transform.
DIVIDE BOTH THE NUMERATOR AND THE DENOMINATOR BY THE MEMBER OF THE DENOMINATOR THAT HAS THE HIGHEST EXPONENT.
First, we figure out which member of the denominator has the highest exponent.
Here is this n2,
but it is under a cube root.
Then, we have this n3,
but it doesn't have a chance, as it is under a 5th radical.
Finally, there is this n,
well, this seems to be the winner.
The member in the denominator that has the highest exponent is n, so we divide by that.
But if we bring it under the radical sign, it goes through some magical changes.
So, under the various radical signs, we divide by different powers of n.
Here come some very funny sequences.
Here is the first one.