1. ## how to go about finding functions from limits?

Hi,

I was wondering if anyone had any thoughts on how to find examples of functions f(x) & g(x) which don't have limits for x -> 1 but do for which f(x) + g(x)?

Not sure where to start?

2. Originally Posted by infinitepersepctives
Hi,

I was wondering if anyone had any thoughts on how to find examples of functions f(x) & g(x) which don't have limits for x -> 1 but do for which f(x) + g(x)?

Not sure where to start?
Try this: $\displaystyle f(x)=\dfrac{1}{x-1}$; $\displaystyle g(x)=\dfrac{1}{1-x}$

This is not the only possible pairing! You can come up with many different examples (regular functions or piecewise functions).

3. Thanks Chris.

Is there a general rule I should follow... in your above example is looks like the denominator of g(x) is inverse to the denominator of f(x). Do you know if this is a general rule or is the answer to this question more of a brute force, try and see approach?

4. No, Chris's whole point is that, just as there exist an infinite number of pairs of numbers, x and y, such that x+ y= 4, so there exist an infinite number of pairs of functions, f and g, such that f(a)+ g(a) is equal to a given number or even such that $\displaystyle \lim_{x\to a} f(x)+ g(x)$ is equal to a given number.

5. Trying to work through some problems without a book (it's on its way and should be here by Monday or Tuesday). Would the only functions that satisfy the above equation be ones where either f(x) or g(x) alone as x ->1 be where one or both of the f(x) or g(x) give a 0 as the denominator and where f(x) + g(x) don't give 0 as a denominator?