Show by means of example that lim/x→a/[f(x)g(x)] may exist

In summary, the existence of lim/x→a/[f(x)g(x)] does not depend on the individual limits of lim/x→a/f(x) and lim/x→a/g(x). This can be shown through examples such as piecewise functions and rational functions, and even functions that are equal to 1 for rational arguments and 0 for irrational arguments. This demonstrates that the statement is valid and can be applied in various contexts.
  • #1
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Show by means of example that lim/x→a/[f(x)g(x)] may exist even though neither lim/x→a/f(x) nor lim/x→a/g(x) exists.

I have tried using examples such as piecewise functions and rational functions, but can never validate the statement.

Any guidance and help would be great.

Thanks.
 
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  • #2
Let f(x) be any crazy function. Let g(x) = 1/f(x).
 
  • #3
mathman said:
Let f(x) be any crazy function. Let g(x) = 1/f(x).

You forgot to insert Arildno's corollary:
"Let f(x) be any crazy function. Let g(x) = 1/f(x). THEN, g(x) is most likely also a crazy function"

Not very useful in this context, of course, but the result is beautiful, nonetheless. :smile:
 
  • #4
MathewsMD said:
Show by means of example that lim/x→a/[f(x)g(x)] may exist even though neither lim/x→a/f(x) nor lim/x→a/g(x) exists.

I have tried using examples such as piecewise functions and rational functions, but can never validate the statement.

Any guidance and help would be great.

Thanks.

Consider the function which is equal to 1 if its argument is rational and 0 otherwise.
 
  • #5
pasmith said:
Consider the function which is equal to 1 if its argument is rational and 0 otherwise.
I presume you mean "let f(x)= 1 if x is rational, 0 if x is irrational.

And then let g(x)= 0 if x is rational, 1 if x is irrational.


fg(x)= 0 for all x so it trivially differentiable.
 

What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It is denoted by the symbol lim and is used to determine the value that a function approaches as its input gets closer and closer to a given value.

What does it mean for a limit to exist?

A limit exists if the value of the function at the given input approaches a finite number as the input gets closer and closer to the given value. In other words, the function does not have any sudden jumps or holes at the given value.

What is the significance of lim/x→a/[f(x)g(x)]?

This notation represents the limit of the quotient of two functions, f(x) and g(x), as x approaches a. It is used to determine if the quotient of the two functions has a finite limit as x approaches a.

Can lim/x→a/[f(x)g(x)] exist even if lim/x→a/f(x) or lim/x→a/g(x) do not exist?

Yes, it is possible for lim/x→a/[f(x)g(x)] to exist while lim/x→a/f(x) or lim/x→a/g(x) do not exist. This is because the existence of a limit for a quotient of two functions depends on the behavior of both functions near the given value, rather than just one of the functions.

How can I show that lim/x→a/[f(x)g(x)] exists?

To show that lim/x→a/[f(x)g(x)] exists, you can use the limit laws and algebraic manipulation to simplify the expression and determine if a finite value can be obtained as x approaches a. Additionally, you can use graphical or numerical methods to visualize and approximate the limit. Finally, you can use the formal definition of a limit to prove its existence.

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