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Plot some points. What are f(0), f(1/4), f(1/2), f(2), f(2.5), etc.?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.
Sorry but I don't get it. Still lost.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.
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.
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 said:I say for (1), the answer is 0.
The answer for (2) is 1.
Yes
For (1), x tends to an integer. Thus, then f(x) = 1.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?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 both cases the limit is 0. (0 is an integer btw).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?
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.
Can you elaborate a little more?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)##..
In that case, it is 0.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)##.
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 said:In that case, it is 0.
You said except for x = 0. I say the limit is 1?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?
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 said:You said except for x = 0. I say the limit is 1?
So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.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..
That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 0?nycmathguy said:So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.
So, f(x) tends to 0 as x-->0.Delta2 said:That's correct. So what conclusion can you make from this? where does f(x) tend to as x tends to 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 said:So, f(x) tends to 0 as x-->0.
Trust me, I plan to journey through calculus l,ll, and lll. We will see limit questions up the wall.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## .
This one is tricky.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?
No. f(x) = 1 if x is an integer, but for all other numbers, f(x) = 0.nycmathguy said:For (1), x tends to an integer. Thus, then f(x) = 1.
Again, no.nycmathguy said:For (1), x tends to 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:For (2), x tends to 0, which is not an integer.
Thus, f(x) = 1.
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).nycmathguy said:So, f(every decimal number you listed) = 0 because decimal numbers are rational and rational numbers are not integers.
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?
Right, but it's not tricky if you understand the idea of what a limit means.nycmathguy said:This one is tricky.
I say the limit is 5.
Ok. There are many more limits coming our way in time. This is just the beginning of the long journey.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.
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.nycmathguy said:Ok. There are many more limits coming our way in time.
Will do.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.
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.
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.
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.
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.
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.