Question

How do you use the limit definition to find the slope of the tangent line to the graph `f(x)=9x-2` at (3,25)?

Answer 1

the slope of the tangent is `9`

Explanation:

The limit definition implies the formula `f'(x) =lim_(h->0) (f(x + h) - f(x))/h`

`f'(x) =lim_(h->0) (9(x + h) - 2 -(9x - 2))/h`

`f'(x) = lim_(h->0) (9x + 9h - 2 - 9x + 2)/h`

`f'(x) = lim_(h->0) (9h)/h`

`f'(x) = lim_(h->0) 9`

`f'(x) = 9`

The graph of this is a horizontal line. The slope of the tangent therefore is always the same, it being `9`.