Limits (coming up with functions that exist and don't exist)

[jules660]

Please help me figure out this question. I'm not sure if I need to come up with a composite function or a piecewise function or what?

 

[Dr.Peterson]

Please help me figure out this question. I'm not sure if I need to come up with a composite function or a piecewise function or what?

It might be any sort of function; you may find that a composite function or a piecewise function seems like a good thing to look for, but that is not immediately clear. I wouldn't start with that question.

What I would tend to do here is to think backward. Let's say that h(x) = f(x)g(x); then f(x) = h(x)/g(x). So (a) becomes:

Find an example of two functions g and h such that lim g(x) and lim h(x) exist, but lim h(x)/g(x) does not exist.​

What might make h(x)/g(x) not have a limit?

 

[HallsofIvy]

For (a) take g(x)= x and f(x)= 1/x, a= 0. \(\displaystyle \lim_{x\to 0} g(x)= \lim_{x\to 0} x= 0\) and \(\displaystyle \lim_{x= 0} g(x)f(x)= \lim_{x\to 0} 1= 1\) but \(\displaystyle \lim_{x\to 0} f(x)\) does not exist.

For (b) reverse those, taking g(x)= 1/x, f(x)= x. Then \(\displaystyle \lim_{x\to 0} f(x)= 0\), \(\displaystyle \lim_{x\to 0} \frac{f(x)}{g(x)}= \lim_{x\to 0} x^2= 0\) but \(\displaystyle \lim_{x\to 0} g(x)\) does not exist.

(It is not a question of the functions "existing or not existing" but the limits.)

 

[jules660]

For (a) take g(x)= x and f(x)= 1/x, a= 0. \(\displaystyle \lim_{x\to 0} g(x)= \lim_{x\to 0} x= 0\) and \(\displaystyle \lim_{x= 0} g(x)f(x)= \lim_{x\to 0} 1= 1\) but \(\displaystyle \lim_{x\to 0} f(x)\) does not exist.

For (b) reverse those, taking g(x)= 1/x, f(x)= x. Then \(\displaystyle \lim_{x\to 0} f(x)= 0\), \(\displaystyle \lim_{x\to 0} \frac{f(x)}{g(x)}= \lim_{x\to 0} x^2= 0\) but \(\displaystyle \lim_{x\to 0} g(x)\) does not exist.

(It is not a question of the functions "existing or not existing" but the limits.)

Thank you so so much!