# Proof of L'Hospital's Rule 3: Lim f(x)/g(x) = ∞

• ocalvino
In summary, the conversation is discussing a proof that if the limit of f(x) as x approaches infinity is equal to infinity, and the limit of g(x) as x approaches infinity is also equal to infinity, then the limit of f(x)/g(x) as x approaches infinity will also equal infinity. The speaker is asking for help understanding the steps of the proof, specifically how to choose a value for k. They also provide additional hints that involve using the Cauchy Mean Value Theorem.
ocalvino
if lim f(x)= infinity= lim g(x)
x->infinity x->infinty

and lim f'(x)/g'(x)=infinity
x-> infinity

then lim f(x)/g(x)=inifity
x-> inifinity

The above fact is what I am trying to prove. From my notes, i see the following:

For m>0, choose k>0, such that if x> k* and g(x),f(x)>0,

then f'(x)/g'(x)> m(4/3).

this is actually where i get lost (so early into the process). can someone explain to me where exactly the prof is headed to with this info? also, is k a functional value through m? if so...how do i choose such k?

i just realized that my post is quite confusing. i suppose I am not too sure what to ask. If you don't understand my original post, perhaps u can just help me start off the proof. thanks.

i also have the following hints to use after the hints in the original post:

fix a>k, then by cauchy mvt,

f(x)-f(a) f'(c)
_______ = ____ > m(4/3) for x>a>k*, and such that
g(x)-g(a) g'(c)

g(x)>g(a)>0 and f(x)> f(a)> 0 where c>a>k*

## 1. What is L'Hospital's Rule?

L'Hospital's Rule is a mathematical tool used to evaluate indeterminate limits of the form 0/0 or ∞/∞. It states that if the limit of the quotient of two functions is indeterminate, then the limit of the quotient of their derivatives will give the same result.

## 2. When is L'Hospital's Rule applicable?

L'Hospital's Rule is applicable when the limit of the quotient of two functions is indeterminate (0/0 or ∞/∞) and the functions are differentiable in a neighborhood of the limit point.

## 3. What is "Proof of L'Hospital's Rule 3"?

"Proof of L'Hospital's Rule 3" refers to the third case of L'Hospital's Rule, which states that if the limit of the quotient of two functions is ∞, then the limit of the quotient of their derivatives will also be ∞. This case is often used when evaluating limits involving exponential functions.

## 4. How do you use L'Hospital's Rule to evaluate a limit?

To use L'Hospital's Rule, you must first check if the limit is indeterminate (0/0 or ∞/∞). If it is, then you can take the derivative of both the numerator and denominator of the quotient and evaluate the limit again. If the limit remains indeterminate, you can repeat this process until the limit is no longer indeterminate or until the derivatives become too complex to evaluate.

## 5. Are there any limitations to using L'Hospital's Rule?

Yes, there are limitations to using L'Hospital's Rule. It can only be used when the limit is indeterminate and the functions are differentiable. It also cannot be used to evaluate limits at infinity or for limits involving oscillating functions. Additionally, it may not always give the correct result if used incorrectly.

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