Question

How do you use the limit definition to find the slope of the tangent line to the graph `F(x) = (( x^2) – 9)` at (2, -5)?

Answer 1

The slope of the tangent line to `f(x)` at `x=h` can be expressed using the limit expression:

`f'(h)=lim_(xrarrc)(f(x)-f(c))/(x-c)`

Here, where `f(x)=x^2-9` and `c=2`, implying that `f(c)=f(2)=-5`, this becomes:

`f'(2)=lim_(xrarr2)((x^2-9)-(-5))/(x-2)`

`f'(2)=lim_(xrarr2)(x^2-4)/(x-2)`

`f'(2)=lim_(xrarr2)((x+2)(x-2))/(x-2)`

`f'(2)=lim_(xrarr2)(x+2)`

`f'(2)=2+2=4`