Question

How do you use the limit definition to find the slope of the tangent line to the graph `f(x)=(3x-1)/(x-1)` at x=0?

Answer 1

Please see below.

Explanation:

The slope of the line tangent to the graph of `f(x)` at `x=a` is

`lim_(xrarra)(f(x)-f(a))/(x-a)`, `" "` if the limit exists.

For this function, the slope of the tangent at `0` is

`lim_(xrarr0)(f(x)-f(0))/(x-0) = lim_(xrarr0) ((3x-1)/(x-1)-1)/x`

`= lim_(xrarr0) ((3x-1)-(x-1))/(x(x-1))`

`= lim_(xrarr0) (2x)/(x(x-1))`

`= lim_(xrarr0) 2/(x-1)`

`= -2`