If ##g## is any bounded function then there is a straightforward epsilon-delta proof.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.
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)).
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.
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.
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.
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.