How do you use the limit definition to find the slope of the tangent line to the graph #F(x) = ((12) / (x - 9)) # at (3,2)?

1 Answer
Jim H
Apr 27, 2016

There is an error in the question. The point #(3,2)# is not on the graph of this function. So there is no tangent line to the graph at that point.

Explanation:

Assuming that the question should read:

Find the slope of the line tangent to the graph of #F(x) = 12/(x-9)# at #(3,-2)#

#F'(x) = lim_(hrarr0)(F(3+h)-F(3))/h#

# = lim_(hrarr0)(12/(3+h-9) - 12/(3-9))/h#

# = lim_(hrarr0)((12(-6) - 12(-6+h))/((-6)(-6+h)))/h#

# = lim_(hrarr0)(-12h)/((-6)(-6+h)h)#

# = lim_(hrarr0)(-12)/((-6)(-6+h))#

# = (-12)/((-6+0)(-6)) = -1/3#

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