Derivative Existence at Vertical Asymptotes: A Mystery or a Certainty?

[you878]

Does a derivative exist at a vertical asymptote of a function?

For the function 1/(x^2), there is a vertical asymptote at x=0. I know that the limit of the function at x=0 is Infinity, but is the derivative at x=0 also infinity, or does it just not exist?

 

[micromass]

Hello you878!

For a function to have a derivative at a point, the function must be defined in that point. The function 1/x², is not defined in 0, hence there is no derivative.

 

[you878]

Thanks for the clarification. I had a feeling it wouldn't exist, but was a little unsure.

 

[HallsofIvy]

The derivative of f(x) at x0 is
lim_(h-> 0) (f(x0+ h)- f(x0))/h

if f(x0) does not exist then then the derivative there does not exist.