Question

How do you find the tangent line of a point on a line?

Answer 1

Because of the way tangent line to a curve is defined, the tangent line to a straight line at any (every) point on the line is the straight line.

Explanation:

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

the line containing `(a,f(a))` with slope `f'(a) = lim_(xrarra)(f(x)-f(a))/(x-a)` if the limit exists.

For a straight line, we have `f(x) = mx+b`
`(a,f(a)) = (a, ma+b)`
and the limit used above turns out to be `m`.

The line through `(a, ma+b)` with slope `m` is `y=mx+b`.

Alternatively

Some presentations use the definition of the slope of the tangent line at the point where `x = a` in the form:

`lim_(hrarr0)(f(a+h)-f(a))/h`.

This form gives the same answer as the one used above.