Question

Use the limit definition to find an equation of a tangent line of f(x)= 1/(x-1) at the given point (2,1)

Answer 1

By "limit definition", I assume you mean the derivative. We have

`f'(c) = \displaystyle\lim_(x\to c)(f(x)-f(c))/(x-c)`

For
`f(x)=\frac1{x-1}` and c = 2, we have

`f'(2)=\displaystyle\lim_(x\to2)\frac{\frac1{x-1}-1}{x-2}`

`f'(2)=\displaystyle\lim_(x\to2)((1-(x-1))/(x-1))/(x-2)`

`f'(2)=\displaystyle\lim_(x\to2)(2-x)/((x-1)(x-2))`

`f'(2)=\displaystyle\lim_(x\to2)\frac1{1-x}=-1`

The tangent line to (2, 1) has slope -1. Using the point-slope formula, it has equation

y - 1 = - (x - 2)

y = - x + 3