Lim x->c[f(x)+g(x)]: Examples to Show No Implication

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In summary, the conversation discusses examples that show that the existence of a limit for the sum or product of two functions does not necessarily imply the existence of a limit for either of the individual functions. The examples of f(x)=1/x and g(x)=-1/x are used to illustrate this point, with f(x)+g(x) still having a limit of 0 even though the individual functions do not. This concept is further demonstrated with the examples of x^2 and 1/x^2, where the limit of the product does not exist even though the individual limits do.
  • #1
ussjt
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Can someone point me in the right dirrection with this problem:

find examples to show that if

a) lim x->c [f(x)+g(x)] exists, this does not imply that either lim x->c [f(x)] or lim x->c [g(x)] exists.

b) lim x->c [f(x)*g(x)] exists, this does not imply that either lim x->c [f(x)] or lim x->c [g(x)] exists.

any help would be great
 
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  • #2
Think simply: if f(x)= 1/x, what is the limit as x-> 0? what is limit of g(x)= -1/x as x-> 0? What about f(x)+ g(x)?
 
  • #3
would that still work for part B of the problem?
 
  • #4
Not that example because there is no limit of f(x)*g(x). Use the two examples of x^2 and 1/x^2
 

1. What does the notation "Lim x->c[f(x)+g(x)]" mean?

The notation "Lim x->c[f(x)+g(x)]" represents the limit of the sum of two functions, f(x) and g(x), as x approaches the value c.

2. What is an example of a function that would show no implication for this limit?

An example of two functions that would show no implication for this limit is f(x) = x and g(x) = -x. In this case, no matter what value c is, the limit of f(x)+g(x) as x approaches c will always be equal to 0.

3. How can I prove that there is no implication for this limit?

To prove that there is no implication for this limit, you can use the epsilon-delta definition of a limit. You will need to show that for any given value of epsilon, there exists a corresponding value of delta that satisfies the definition. If you can find two different functions that satisfy this condition, then it can be concluded that there is no implication for the limit.

4. Can the limit of f(x)+g(x) be equal to a number other than 0?

Yes, the limit of f(x)+g(x) can be equal to any real number other than 0. This will depend on the individual functions f(x) and g(x) and the value of c.

5. What other types of implications can be shown using limits?

Other types of implications that can be shown using limits include implication for sums, products, quotients, and compositions of functions. These implications can be used to prove various mathematical theorems and properties.

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