Can anyone solve 1st and 2nd questions? http://www.fen.bilkent.edu.tr/~otekman/calc1/quiz/Quiz1,Sec03.pdf
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.
You should upgrade or use an alternative browser.
2 limit questions: Does lim, x->0, (x*f(1/x)) exist?
- Thread starter berk
- Start date
I'm sure plenty of people can, but the real question is: can you solve them? I'll type out the exercises for any of the other helpers here who might not be willing to go to a random link:
So, what have you tried on these problems? Please share with us any and all work you've done on these problems, even the parts you know for sure are wrong. Thank you.
EDIT: Correcting my mis-reading of the handwritten problem.
1) Determine whether the statements are true or false. If they are true, explain why. If they are false, explain why or give an example that disproves the statement.
a) \(\displaystyle \displaystyle \lim _{x\to 0}\left(x\cdot f\left(\frac{1}{x}\right)\right)\) exists
b) \(\displaystyle \displaystyle \text{If } -x^2 < f(x) < x^2 \: \: \: \forall x \ne 0, \lim _{x\to 0}\left(f\left(x\right)\right)=0\)
2) Find \(\displaystyle \displaystyle \lim _{x\to 2}\left(\frac{\sqrt{6-x}-2}{\sqrt[3]{3-x}-1}\right)\)
So, what have you tried on these problems? Please share with us any and all work you've done on these problems, even the parts you know for sure are wrong. Thank you.
EDIT: Correcting my mis-reading of the handwritten problem.
Last edited:
I solved the 2nd question by letting y^6=3-x. But I still don't have any idea about 1-a. As for 1-b I think I would solve this question if they told us f(x) is a continuous function but we don't know that it is a continuous function or not. By the way 1-b is -x^2 I think you have made a mistake while writing the questionI'm sure plenty of people can, but the real question is: can you solve them? I'll type out the exercises for any of the other helpers here who might not be willing to go to a random link:
So, what have you tried on these problems? Please share with us any and all work you've done on these problems, even the parts you know for sure are wrong. Thank you.
I solved the 2nd question by letting y^6=3-x. But I still don't have any idea about 1-a. As for 1-b I think I would solve this question if they told us f(x) is a continuous function but we don't know that it is a continuous function or not. By the way 1-b is -x^2 I think you have made a mistake while writing the question
Ah, yes, I did misread the handwritten problem. I've corrected it. Thank you. So, now, let's think carefully about what the two parts of problem 1 are asking. Part (a) sets up a limit and asks us if it exists. We don't know what the function f(x) is though. It could be any function we like, including f(x)=7 or f(x)=x. If we could come up with even one example of an f(x) where it's not true that would prove it's false (remember that for a statement to be considered "true" in math, it must be *always* true. Statements that are sometimes true are considered "false"), but the reverse is not true - that is, no matter how many functions we find it's true for, there could still always be one where it's not. In this case, I'll tell you that the limit doesn't always exist, so save you from going down the rabbit hole of a delta-epsilon proof, although I'll leave it to you to find an example where it's not true, and figure out what properties of f(x) would make it not true.
Some things you might consider are: What are the criteria for a limit to exist? What, then, does it mean for a limit to not exist? In the context of the problem, what happens if f(1/x) is undefined at the point x=0? What happens if x * f(1/x) is undefined at the point x=0? Recall that sometimes the limit as x approaches a target value from below will be different than the limit as x approaches that same target value from above. What would it mean if this were true of the given limit?
For part (b), you're given an inequality and told that if this inequality holds from all non-zero values of x, then the limit of that function as x approaches 0 will be 0. Since the "then" part of the if-then statement is that the limit is 0, a good place to start might to find a function whose limit as x approaches 0 isn't 0, and work backwards. During your brainstorming for part (a) you most likely found plenty of functions where the limit isn't 0 because it doesn't exist. What properties do those functions have? What does that mean about the inequality? Maybe also find some functions whose limit does exist but is a non-zero real number. What properties do those functions have? What does that mean about the inequality?