L'Hospital's Rule

[ahorn]

L'Hospital's Rule, which states that:
lim_x->a[f(x)/g(x)] = lim_x->a[f'(x)/g'(x)]

can be applied when:
(1) f, g are differentiable
(2) g'(z) ≠ 0 for z near a, except possibly at a
(3) lim_x->a[f(x)] = 0 = lim_x->a[g(x)], or
lim_x->a[f(x)] = ±∞ = lim_x->a[g(x)] ,
(4) if the limit on the RHS exists or if it equals ±∞

My question is:

Why is proviso (2) necessary?

 

[stapel]

L'Hospital's Rule, which states that:
lim_x->a[f(x)/g(x)] = lim_x->a[f'(x)/g'(x)]

can be applied when:
(2) g'(z) ≠ 0 for z near a...

Why is proviso (2) necessary?

Can you divide by zero?

 

[ahorn]

Can you divide by zero?

Yes, and the answer will be \(\displaystyle \infty\).
e.g. \(\displaystyle \displaystyle\lim_{x \to 0}\frac{x}{1-cosx}=\displaystyle\lim_{x \to 0}\frac{1}{sinx}=\frac{1}{0}=\infty\)

 

[stapel]

Yes, and the answer will be \(\displaystyle \infty\).
e.g. \(\displaystyle \displaystyle\lim_{x \to 0}\frac{x}{1-cosx}=\displaystyle\lim_{x \to 0}\frac{1}{sinx}=\frac{1}{0}=\infty\)

Taking the limit as a value by which one is dividing "tends toward" zero is not the same as actually dividing by zero!

 

[ahorn]

My maths teacher just told me that it is because you cannot evaluate g'(z) when g'(z)=0 near a. So, why does the gradient near a point affect the gradient at that point? Is it because the gradient does not exist at that point? Is there somewhere I can read about the original thought behind this? Like a book or website?