]]>

**Conversation**

*Smith*: Ah. The 2h results from the subtraction of h from – f(x)?

*Alam*: Using the rule of the hospital with dy/dh instead of dy/dx: lim [h to 0] of ( f’(x+h) – -f’(x-h) ) /(2). As h –> 0, the equation –> ( 2f’(x)/2) = f’(x).

20120214

]]>If , then evaluate the limit rewritten under a common denominator.

If , then the limit is equivalent to:

20120214

]]>and

Also note that

]]>On the same interval, the instantaneous rate of change is given by the slope of the tangent line:

It is also noted that

when qualified by . Therefore, the two definitions are conceptually same.

When I tried to understand differential calculus prior, I could not get past this concept. But now I completely understand it:

where and Therefore, the derivative of a function is defined as:

]]>however, it seems to be a restatement of another limit theorem:

Here is my attempt to generalize equation (1) in terms of equation (2):

]]>If the functions and have limits and that exist, then it is true that and exist. For justification, consider:

and

It is apparent that the limits of both functions as exist:

and

Since these limits exist, it follows that:

and

**III-B**

Consider the functions

and

It is apparent the the limit of these functions as do not exist. However:

Now consider the functions

and

It is apparent that the limits of these functions as do not exist; however:

]]>Given that the continuous functions and are rational, and that and exist, it follows that and

**Question III-A**

Given that and the operation is defined for all real numbers, it follows that the limit of the sum of the functions is also defined for all real numbers:

** Question III-B**

Given that the functions and are defined for all real numbers and that and do *not* exist, it follows that and do not exist as well:

and

]]>or

For the same function, there exists a horizontal asymptote for a constant * *if

or

]]>