let fx = 2 / (3-x)

Question

(a)Find the slope of the tangent to the graph of f at a general point x0 using the definition
of limits.
(b) Use the result in part (a) to find the slope of the tangent line at x0 = 1:

Answer 1

The slope of the tangent line is the derivative of function f(x). 

 

 

Part a)

 

Using the definition of limits.

 

lim          f(x + h) - f(x)

h-->0     ___________

                    h

 

 

lim       (2 / [3 - (x + h)]) - (2 / (3 - x))

h-->0  ___________________________

                             h

 

 

lim       (2 / (3 - x - h)) - (2 / (3 - x))

h-->0  ____________________________

                           h

 

 

The LCD in the numerator part is   (3 - x)(3 - x - h).

 

 

lim        [(2(3 - x)) / ((3 - x)(3 - x - h))] - [(2(3 - x - h)) / ((3 - x)(3 - x - h))]

h-->0   ____________________________________________________________

                                                          h

 

 

lim        (6 - 2x - 6 + 2x + 2h) / ((3 - x)(3 - x - h))

h-->0   _________________________________________

                                      h

 

 

lim        (2h / (9 - 3x - 3h - 3x + x² + xh))

h-->0   __________________________________

                                     h

 

 

The green terms cancel out.

 

 

lim                            2

h-->0      _____________________________

                     9 - 3x - 3h - 3x + x² + xh

 

 

The h terms cancel out.

 

 

lim                          2

h-->0    __________________________

                     9 - 6x + x²

 

 

This is the derivative of f(x).  Now, just plug in x=0.

 

 

f'(0) = 2 / 9

 

 

The slope of the tangent line at f(x) is 2/9.

 

 

Part b)

 

Plug in x=1 into the derivative.

 

f'(1) = 2 / (9 - 6(1) + 1²)

       = 2 / (9 - 6 + 1)

       = 2 / 4

       = 1 / 2

 

The slope of the tangent line at f(x) is 1/2.