How Do Limits Behave for Piecewise Functions at Specific Points?

  • Thread starter nycmathguy
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I am a math major.In summary, the conversation focused on investigating the limits of a piecewise function with the definition f(x)=1 if x is an integer and f(x)=0 if x is not an integer. The experts discussed how the function behaves at different intervals and how to determine the limit as x approaches different values, including integers and rational numbers. They also discussed how the function can be tweaked by giving a specific value to f(0) and how this affects the limit. The conclusion was that regardless of the value given to f(0), the limit of the function at x=0 will always be 0 as long as f(x)=0 for all x except 0.
  • #1
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.
 

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  • #2
nycmathguy said:
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.
 
  • #3
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.
 
  • #4
Delta2 said:
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.
 
  • #5
what is f(1) and f(2) equal to for example? Hint: 1 and 2 are integers
 
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  • #6
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.
 
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  • #7
Delta2 said:
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?
 
  • #8
nycmathguy said:
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.
 
  • #9
Delta2 said:
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.
 
  • #10
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.
 
  • #11
Delta2 said:
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?
 
  • #12
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)##..
 
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  • #13
nycmathguy said:
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).
 
  • #14
nycmathguy said:
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.

Delta2 said:
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.
 
  • #15
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)##.
 
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  • #16
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.
 
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  • #17
Delta2 said:
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.
 
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  • #18
nycmathguy said:
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?
 
  • #19
Delta2 said:
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?
 
  • #20
nycmathguy said:
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..
 
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  • #21
Delta2 said:
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.
 
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  • #22
nycmathguy said:
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?
 
  • #23
Delta2 said:
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.
 
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  • #24
nycmathguy said:
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## .
 
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  • #25
Delta2 said:
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.
 
  • #26
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?:wink:
 
  • #27
Delta2 said:
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?:wink:
This one is tricky.
I say the limit is 5.
 
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  • #28
nycmathguy said:
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.
nycmathguy said:
For (1), x tends to an integer. Thus, f(x) = 1.
Again, no.
nycmathguy said:
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).
nycmathguy said:
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).

Delta2 said:
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?

nycmathguy said:
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.
 
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  • #29
Mark44 said:
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.
 
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  • #30
nycmathguy said:
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.
 
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  • #31
Mark44 said:
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.
Will do.
 
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Related to How Do Limits Behave for Piecewise Functions at Specific Points?

What is a limit in mathematics?

A limit in mathematics is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a particular value.

How do you investigate a limit?

To investigate a limit, you need to evaluate the function at values that are closer and closer to the value the input is approaching. You can also use algebraic techniques, such as factoring and simplifying, to manipulate the function and determine its limit.

What is the purpose of investigating limits?

Investigating limits helps us understand the behavior of a function and its values as the input approaches a particular value. It is also used to determine if a function is continuous at a certain point, which is important in many real-world applications.

What are the different types of limits?

The two main types of limits are one-sided limits and two-sided limits. One-sided limits only consider the behavior of a function as the input approaches from one side, while two-sided limits consider the behavior from both sides of the input value.

What are some common techniques for evaluating limits?

Some common techniques for evaluating limits include direct substitution, factoring, rationalizing the numerator or denominator, and using trigonometric identities. You can also use L'Hospital's rule, which involves taking the derivative of the numerator and denominator separately.

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