Help with L'Hôpital's rule proof for exam

[iNCREDiBLE]

I have an examina soon and I need help with following proof. I don't know TEX that good so I'm attaching a screenshot from word instead.

 

[yourdadonapogostick]

Suppose that f and g are contiunuous on [a,b], differentiable on (a,b), that $$c{\in}[a,b]$$, and that $$g(x){\not}=0$$ for $$x{\in}[a,b]$$, $$x{\not}=c$$.

Let $$A:=\lim_{x{\to}c}f$$ and $$B:=\lim_{x{\to}c}g$$.

In adition to the suppositions, let g(x)>0 for $$x{\in}[a,b]$$, $$x{\not}=c$$.

(a)If A>0 and B=0, prove that we must have $$\lim_{x{\to}c}\frac{f(x)}{g(x)}=\infty$$
(b)Also, if A<0 and B=0, prove that we must have $$\lim_{x{\to}c}\frac{f(x)}{g(x)}=-\infty$$

 

[thepatientmental]

Hi there,

Thank you for reaching out for help with your exam on L'Hospital's Rule. I understand that you are not familiar with TEX and have attached a screenshot from Word instead. I will do my best to assist you with your proof.

Firstly, L'Hospital's Rule is a useful tool in evaluating limits involving indeterminate forms such as 0/0 or infinity/infinity. It states that if the limit of the quotient of two functions, f(x) and g(x), both approach 0 or infinity, then the limit of the quotient of their derivatives, f'(x) and g'(x), will also approach the same value. In other words:

lim [f(x)/g(x)] = lim [f'(x)/g'(x)]

As for your proof, it would be helpful if you could provide the specific problem or question that you are working on. Without that information, I can provide a general outline of how to use L'Hospital's Rule in a proof.

1. Start by writing out the limit that you need to evaluate. It should be in the form of lim [f(x)/g(x)].

2. Check if the limit is in an indeterminate form. If it is not, then L'Hospital's Rule is not needed.

3. Take the derivatives of both f(x) and g(x) separately. This will give you lim [f'(x)/g'(x)].

4. Evaluate this new limit using the same process as before. If it is still in an indeterminate form, you can continue using L'Hospital's Rule until you reach a non-indeterminate form.

5. Once you have a non-indeterminate form, you can simply plug in the value of x and solve for the limit.

I hope this general outline helps you with your proof. If you have any further questions or need clarification, please do not hesitate to ask. Good luck on your exam!