2 functions f and g that dont have limits at a number c but fg and f+g do

In summary, the conversation discusses the limits of two functions, f and g, at a real number c. It is mentioned that if both f and g have limits at c, then the limits of f+g and fg can be determined. However, the possibility of f and g not having limits at c is also raised, with an example provided where f=-g=1/x. In this case, the limit of f+g is 0 while the limit of fg is -infinity.
  • #1
ibensous
10
0
I was wondering if anyone knows of an example where f and g are two functions that do not have limits at the real number c but f+g and fg have limits at c.

I know that if f and g are functions and L= limx->c f(x) and D = limx->c g(x) then the limx->c (f+g) = L + D and limx->c (fg) = LD but that's assuming both L and D exist. What if L and D don't exist?
 
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  • #2
What if f = -g = 1/x?

lim x--> 0 of f or g is undefined, but lim x--> 0 f + g = 0

lim x--> 0 fg would be - infinity.
 

FAQ: 2 functions f and g that dont have limits at a number c but fg and f+g do

1. What does it mean for a function to have a limit at a specific number?

A limit is a mathematical concept that describes the behavior of a function as the input approaches a specific value. It represents the value that the function is "approaching" as the input gets closer and closer to the specified value. A function has a limit at a number c if the value of the function approaches a single number as the input gets closer and closer to c.

2. Can a function have a limit at a number c but its product with another function does not?

Yes, it is possible for a function to have a limit at a number c, but its product with another function does not. This means that the individual functions may have different behaviors as the input approaches c, but when multiplied together, their product does not have a defined limit at c.

3. How does the limit of a sum of two functions differ from the sum of their individual limits?

The limit of a sum of two functions can be different from the sum of their individual limits. This is because the limit of a sum is the sum of the individual limits only if both individual limits exist. If one or both individual limits do not exist, the limit of the sum cannot be determined solely from the individual limits.

4. Why do functions f and g need to have limits at a number c for their product and sum to have limits at c?

This is because the behavior of a function at a specific value is dependent on the behavior of the individual functions that make up the product or sum. If the individual functions do not have limits at c, the product or sum cannot have a defined limit at c.

5. Can a function have a limit at a number c but not exist at that number?

Yes, it is possible for a function to have a limit at a number c but not exist at that number. This means that the function may have a defined limit as the input approaches c, but the function itself is not defined at c. This can happen if there is a hole or gap in the function at c, or if the function is not defined at c due to a vertical asymptote.

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