Limit of the product of these two functions

In summary, the discussion was about two functions ##f(x)## and ##g(x)## with certain limits at infinity. The question was whether it is true that ##\lim_{x \to \infty}f(x)g(x)=0## and if there is a theorem or book that clearly states this. The answer is that this is an application of the Squeeze theorem, which can be found in any calculus textbook.
  • #1
LagrangeEuler
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If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that
[tex]\lim_{x \to \infty}f(x)g(x)=0[/tex]
I found that only for sequences, but it should be correct for functions also.
 
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  • #2
LagrangeEuler said:
If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that
[tex]\lim_{x \to \infty}f(x)g(x)=0[/tex]
I found that only for sequences, but it should be correct for functions also.
If ##g## is any bounded function then there is a straightforward epsilon-delta proof.
 
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Likes Svein
  • #3
|f(x)g(x)| is bounded by |f(x)| goes to 0
 
  • #4
I don't know of a specific place that states this exact problem, but this is an application of the Squeeze theorem which can be found in any calculus textbook.
 

1. What is the definition of a limit of the product of two functions?

The limit of the product of two functions is the value that the product approaches as the independent variable (usually denoted as x) approaches a specific value or "approaches" infinity or negative infinity. It is denoted by the notation lim f(x) * g(x) or lim (f(x) * g(x)).

2. How is the limit of the product of two functions different from the limit of a single function?

The limit of a single function is the value that the function approaches as the independent variable approaches a specific value or infinity. The limit of the product of two functions involves finding the limit of the individual functions and then multiplying them together to find the overall limit.

3. Can the limit of the product of two functions exist if the individual limits do not exist?

Yes, it is possible for the limit of the product of two functions to exist even if the individual limits do not exist. This can happen if the product of the two functions approaches a finite value or if it oscillates between two values as the independent variable approaches a specific value.

4. What are some common techniques for finding the limit of the product of two functions?

Some common techniques for finding the limit of the product of two functions include using algebraic manipulation, factoring, and using L'Hopital's rule. It is also important to consider any special cases, such as when the product involves a trigonometric function or a rational function.

5. How can the limit of the product of two functions be used in real-world applications?

The limit of the product of two functions can be used in various real-world applications, such as in physics, economics, and engineering. For example, it can be used to determine the maximum profit for a company by finding the limit of the product of the revenue and cost functions. It can also be used to calculate the velocity of an object by finding the limit of the product of its displacement and velocity functions.

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