Question: "How do you use the limit definition to..."

Esther Hoffman

Answered question

2022-04-22

How do you use the limit definition to find the slope of the tangent line to the graph

F (x) = (\frac{12}{x - 9})

at (3,2)?

Leanna Boone

Beginner2022-04-23Added 11 answers

Assuming that the question should read:
Find the slope of the line tangent to the graph of

F (x) = \frac{12}{x - 9}

at (3, -2)

F^{'} (x) = lim_{h \to 0} \frac{F (3 + h) - F (3)}{h}

= lim_{h \to 0} \frac{\frac{12}{3 + h - 9} - \frac{12}{3 - 9}}{h}

= lim_{h \to 0} \frac{\frac{12 (- 6) - 12 (- 6 + h)}{(- 6) (- 6 + h)}}{h}

= lim_{h \to 0} \frac{- 12 h}{(- 6) (- 6 + h) h}

= lim_{h \to 0} \frac{- 12}{(- 6) (- 6 + h)}

= \frac{- 12}{(- 6 + 0) (- 6)} = - \frac{1}{3}

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