How Do Limits Behave for Piecewise Functions at Specific Points?

[nycmathguy]

Homework Statement

Investigate each limit.

Relevant Equations

See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

 

[Mark44]

Homework Statement:: Investigate each limit.
Relevant Equations:: See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

Plot some points. What are f(0), f(1/4), f(1/2), f(2), f(2.5), etc.?

Also, you've posted a few threads just now with little or no work shown. That's a violation of forum rules. You have to show some effort. You have the formula for the function -- sketch a graph of it.

 

[Delta2]

Focus at an interval [n,n+1] where n is an integer. Answer the following questions to help you understand how this function goes
1) What is f(n)
2) What is f(n+1)
3) What is f(x) for every $x\in(n,n+1)$ for example for x=(2n+1)/2 the midpoint of n and n+1.

 

[nycmathguy]

Focus at an interval [n,n+1] where n is an integer. Answer the following questions to help you understand how this function goes
1) What is f(n)
2) What is f(n+1)
3) What is f(x) for every $x\in(n,n+1)$ for example for x=(2n+1)/2 the midpoint of n and n+1.

Sorry but I don't get it. Still lost.

 

[Delta2]

what is f(1) and f(2) equal to for example? Hint: 1 and 2 are integers

 

[Delta2]

The definition of the function f(x) tells you that f(x)=1 if x is integer and f(x)=0 if x is not integer.

 

[nycmathguy]

The definition of the function f(x) tells you that f(x)=1 if x is integer and f(x)=0 if x is not integer.

I say for (1), the answer is 0.
The answer for (2) is 1.

Yes?

 

[Delta2]

I say for (1), the answer is 0.
The answer for (2) is 1.

Yes

Νο, $f(n)=f(n+1)=1$ for all integers n. The function definition tells us that f(x)=1 if x is integer.

 

[nycmathguy]

Νο, $f(n)=f(n+1)=1$ for all integers n. The function definition tells us that f(x)=1 if x is integer.

For (1), x tends to an integer. Thus, then f(x) = 1.

For (2), x tends to a rational number. Thus, f(x) = 0.

 

[Delta2]

Yes but as x tends to an integer, it passes from all sorts of rationals and irrationals (from the left and right of integer) for which f(x)=0.

 

[nycmathguy]

Νο, $f(n)=f(n+1)=1$ for all integers n. The function definition tells us that f(x)=1 if x is integer.

What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.

Yes?

 

[Delta2]

If I give you the following definition for f:
f(1)=f(0)=1
f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the $\lim_{x\to 0} f(x)$ (or $\lim_{x\to 1} f(x)$..

 

[Delta2]

What about the following two cases using the same attachment?

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

For (1), x tends to an integer. Thus, f(x) = 1.

For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.

Yes?

For both cases the limit is 0. (0 is an integer btw).

 

[nycmathguy]

Homework Statement:: Investigate each limit.
Relevant Equations:: See attachment for function.

Investigate each limit.

See attachment.

1. lim f(x) x→2

2. lim f(x) x→1/2

I don't understand this piecewise function.

If I give you the following definition for f:
f(1)=f(0)=1
f(x)=0 for all x inbetween 0 and 1.

Then what do you think is the $\lim_{x\to 0} f(x)$ (or $\lim_{x\to 1} f(x)$..

Can you elaborate a little more?
It's just not sinking in. In fact, Sullivan stated in his book that this is considered a challenging problem.

 

[Delta2]

hm ok let me see
If I tell you that f(x)=0 for all x then what is the $\lim_{x\to 0} f(x)$.

 

[Delta2]

I think you are confusing the $\lim_{x\to x_0}f(x)$ with the $f(x_0)$. These two are equal only if the function f is continuous at $x_0$. But in this problem here we have to deal with a function f that is not continuous at every integer.

 

[nycmathguy]

hm ok let me see
If I tell you that f(x)=0 for all x then what is the $\lim_{x\to 0} f(x)$.

In that case, it is 0.

 

[Delta2]

In that case, it is 0.

Correct now let's say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?

 

[nycmathguy]

Correct now let's say I tweak the function and the function f is now f(x)=0 for all x EXCEPT for x=0 which I define to be f(0)=1. Do you think that the above limit changes or remains the same?

You said except for x = 0. I say the limit is 1?

 

[Delta2]

You said except for x = 0. I say the limit is 1?

Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..

 

[nycmathguy]

Nope it isn't 1. What is f(x) equal to ,as x tends to 0, for example what is f(0.5), f(0.4), f(0.3) , f(0.2) and so on..

So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

 

[Delta2]

So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?

 

[nycmathguy]

That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?

So, f(x) tends to 0 as x-->0.

 

[Delta2]

So, f(x) tends to 0 as x-->0.

yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all $x\neq 0$ .

 

[nycmathguy]

yes and this is true regardless of what value we choose to give to f(0). As long as f(x)=0 for all $x\neq 0$ .

Trust me, I plan to journey through calculus l,ll, and lll. We will see limit questions up the wall.

 

[Delta2]

Just to check your understanding, if i tell you f(x)=5 for all $x\neq 0$ and f(0)=10, what is the limit of f(x) as x tends to 0?

 

[nycmathguy]

Just to check your understanding, if i tell you f(x)=5 for all $x\neq 0$ and f(0)=10, what is the limit of f(x) as x tends to 0?

This one is tricky.
I say the limit is 5.

 

[Mark44]

For (1), x tends to an integer. Thus, then f(x) = 1.

No. f(x) = 1 if x is an integer, but for all other numbers, f(x) = 0.
The question is asking about $\lim_{x \to 2} f(x)$, not f(x). Even though f(2) = 1, $\lim_{x \to 2} f(x)$ is some other value.

For (1), x tends to an integer. Thus, f(x) = 1.

Again, no.

For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.

First off, 0 is an integer. Second, you're again not distinguishing between function values (e.g. f(0)) and values of the limit. Here the limit expression is $\lim_{x \to 1/2} f(x)$, which just happens to be the same as f(1/2).

So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.

Most "decimal" numbers are not rational (e.g., $\pi \approx 3.141592$ and $\sqrt 2 \approx 1.414$), and some rational numbers are integers (e.g., 2/1, 6/2, and so on).

Just to check your understanding, if i tell you f(x)=5 for all x≠0 and f(0)=10, what is the limit of f(x) as x tends to 0?

This one is tricky.
I say the limit is 5.

Right, but it's not tricky if you understand the idea of what a limit means.

 

[nycmathguy]

No. f(x) = 1 if x is an integer, but for all other numbers, f(x) = 0.
The question is asking about $\lim_{x \to 2} f(x)$, not f(x). Even though f(2) = 1, $\lim_{x \to 2} f(x)$ is some other value.

Again, no.

First off, 0 is an integer. Second, you're again not distinguishing between function values (e.g. f(0)) and values of the limit. Here the limit expression is $\lim_{x \to 1/2} f(x)$, which just happens to be the same as f(1/2).

Most "decimal" numbers are not rational (e.g., $\pi \approx 3.141592$ and $\sqrt 2 \approx 1.414$), and some rational numbers are integers (e.g., 2/1, 6/2, and so on).
Right, but it's not tricky if you understand the idea of what a limit means.

Ok. There are many more limits coming our way in time. This is just the beginning of the long journey.

 

[Mark44]

Ok. There are many more limits coming our way in time.

So make sure you understand the difference between, say, $f(c)$ and $\lim_{x \to c} f(x)$. For a continuous function f, they will be the same, but not necessarily so for discontinuous or piecewise-defined functions.