Proving "If...Then" Limit Statement: Help Needed!

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In summary, the conversation is about using L'Hospital's rule to prove a mathematical statement involving limits and derivatives. The person is seeking help on where to begin with the proof. A possible proof using Cauchy's mean value theorem can be found on Wikipedia.
  • #1
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?
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  • #3

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

1. What is a "Proving "If...Then" Limit Statement?

A "Proving "If...Then" Limit Statement is a mathematical statement that describes the relationship between an input value and an output value, and how the output value behaves as the input value approaches a certain value or limit.

2. Why is it important to prove "If...Then" Limit Statements?

Proving "If...Then" Limit Statements is important because it allows us to understand and predict the behavior of mathematical functions and expressions, and to make accurate conclusions about their limits.

3. What are the steps to prove an "If...Then" Limit Statement?

The steps to prove an "If...Then" Limit Statement include: setting up the problem, determining the limit value, finding the "delta" value, and using the "epsilon-delta" definition to prove the statement.

4. What is the "epsilon-delta" definition used in proving "If...Then" Limit Statements?

The "epsilon-delta" definition is a mathematical principle that states that for any given "epsilon" value (a small positive number), there exists a corresponding "delta" value (a small positive number) such that the difference between the input value and the limit value is less than "epsilon". This is used to prove that the limit statement is true.

5. How can I check my work when proving "If...Then" Limit Statements?

You can check your work by using different methods such as graphing the function, using numerical approximations, or consulting a math expert. It is important to double-check your work to ensure accuracy and minimize errors.

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