Question

How do you use the limit definition to find the slope of the tangent line to the graph `f(x) = 4- x^2` at (2,0)?

Answer 1

The slope of the tangent line is `-4`.

Explanation:

When dealing with "limit definition" problems, use the formula `f'(x) = lim_(h->0) (f(x + h) - f(x))/h` to find the derivative.

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

`f'(x) = lim_(h->0) (4 - (x + h)^2 - (4 - x^2))/h`

`f'(x) = lim_(h->0) (4 - x^2 - 2xh - h^2 - 4 + x^2)/h`

`f'(x)= lim_(h-0) (-2xh + h^2)/h`

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

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

`f'(x) = -2x + 0`

`f'(x) = -2x`

We can now substitute the `x` coordinate of your point into the derivative to find the slope the tangent line.

`f'(x) = -2(2)`

`f'(x) = -4`

Hence, the slope of the tangent line is `-4`.

Hopefully this helps!