Also known as Hysterical Calculus

definition of limit to obtain a sandwich inequality for an.
Solution: Since the limit of (2n – 1) an is 16 we have:

Set then

That is,

conditions x1 = a, x2 = b.
Find

Solution. We begin with finding an explicit expression for the general term of the sequence xn.
Let us try the following formula:

defined by the relationship

Divide both sides by to obtain
or

sequences satisfy the initial conditions x1 = a, x2 = b?
Well, if a = b, then the first sequence with c1 = a, satisfies the initial conditions.
If b = – ½ a, then the second sequence with c2 = –2 a, satisfies the initial conditions.
But what should we do if a and b are arbitrary?

defining relationship

and

linear combination of the two obtained sequences

indeed satisfies the equation

the limit is not difficult to find:

Now all we have to do is to find the values of c1 and c2 such that our sequence also satisfies the initial conditions:

defining

relationship

and the initial conditions x1 = a, x2 = b we have to:
1. Write down the characteristic equation

and obtain its roots

2. Write down the general formula for xn:

and find the values of constants c1 and c2, such that x1 = a, x2 = b.

a continuous function we obtain

Hence, the sandwich theorem tells us that

We begin with the definition of a convergent sequence.
A sequence {xn} converges to a number L, if

A sequence {xn} does not converges to a number L, if

A sequence {xn} is divergent, if it does not converges to any number L.

– plane.
Solution. To begin with, we calculate the limit

in the particular case x = – 7, y = 5.

Hence,

equation

We have

Since

the sandwich theorem

tells us that

draw the curve defined by the equation

sandwich inequality for the sequence

Solution: The sequence

and find the limit of this sequence.

converges to 0.
Therefore, according to the definition of the limit,

Choose

and denote N1 – the

corresponding value of N.

inequality for our sequence xn

for all