How Do I Prove This Limit Statement in Calculus?

[ocalvino]

i originally posted this in the analasys section. Since i haven't been getting any results, I've figured that perhaps people feel it should be in the homework section. anyway, I am trying to prove the folloting:

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

To be honest, i don't know where to begin, and that's where I need your help. How do i start to prove this?

 

[Berislav]

The proof using Cauchy's mean value theorem can be found here:

http://en.wikipedia.org/wiki/L'hospital

 

[thepatientmental]

First of all, it is important to understand the concept of a limit. A limit is the value that a function approaches as the input approaches a certain value. In this case, we are dealing with limits as the input (x) approaches infinity.

To prove the statement, we need to use the definition of a limit. According to the definition, if the limit of a function f(x) as x approaches infinity is L, then for any positive number ε, there exists a number N such that for all x greater than N, the difference between f(x) and L is less than ε.

Now, let's start with the given information. We know that lim f(x) = infinity = lim g(x) as x approaches infinity. This means that as x gets larger and larger, both f(x) and g(x) also get larger and larger. In other words, both f(x) and g(x) have no upper bound as x approaches infinity.

Next, we are given that lim f'(x)/g'(x) = infinity as x approaches infinity. This means that the derivative of f(x) divided by the derivative of g(x) approaches infinity as x approaches infinity. In other words, the rate of change of f(x) is greater than the rate of change of g(x) as x approaches infinity.

Now, let's consider the limit of f(x)/g(x) as x approaches infinity. We can rewrite this expression as (f(x)/g(x))*(g(x)/g(x)). Using the quotient rule for derivatives, we can rewrite the expression as (f'(x)*g(x)-f(x)*g'(x))/g(x)^2.

Since we know that both f(x) and g(x) approach infinity as x approaches infinity, we can replace f(x) and g(x) with infinity in the above expression. This gives us (infinity*infinity-infinity*infinity)/infinity^2 = infinity/infinity.

However, we also know that the rate of change of f(x) is greater than the rate of change of g(x) as x approaches infinity. This means that the numerator (f'(x)*g(x)-f(x)*g'(x)) will always be greater than or equal to 0. Therefore, the expression (f'(x)*g(x)-f(x)*g'(x))/g(x)^2 will always be greater than or equal to 0.

Since we have shown