LIMITS

 

 

Rate of change: 
Change in dependent variable with respect to change in independent variable.

 

In the function of y = f (x)

x is independent variable, y is the dependent variable.

 

Slope: 

 

 

 

   

Slope of secant:

Average rate of change of y with respect to x over the interval from x = a to x = a + h

 

 

Slope of tangent:
The slope of the tangent to the graph y = f (x) at point P (a, f(a))
 

 

 

If the slope of a line equal to m,
and then the slope of the line perpendicular to this line will be

 

 

 

Limits:

 

Left hand limit:
   
Right hand limit: 
   
Limit:

 

 

If

then there is a limit.

     
If

then limit does not exist

 

 

 

Continuity

 

A function is continuous at a point when the graph passes through the point without a break.  If the graph has a break of some type at a point, then the graph is not continuous (discontinuous) at this point.

 

The function f (x) is continuous at x = a

 

If   = f (a)

 

 

 

Polynomial Functions:

 

P(x) = anxn + an-1xn-1 + ...a1x + a0

 

a0, a1, a2...an are constant.

All the polynomial functions are have limits at all real values.

All the polynomial functions are continuous at all real values.

 

Rational Functions:

 

      Every rational functions are continuous except at a value:  Q(a) = 0