Evaluating Limits: Explaining Why Limit of f(x) Does Not Exist at x → 1

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In summary: So, the limit from the left does not exist. However, the limit from the right does exist- 1/5, 1/4, 1/3, 1/2, and 1.
  • #1
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Homework Statement



Consider a function f: D∈R, where D = {1/n for natural numbers n (1, 2, 3, 4, etc.)} and f(x) = 3x - 1 for all x in D. Explain why the limit of f(x) as x → 1 does not exist.

Homework Equations





The Attempt at a Solution



Uh I figured it would exist. We know a function does not have a limit at c if and only if there exists a sequence (x_n) where x_n ≠ c for all natural numbers n such that (x_n) converges to c but the sequence (f(x_n)) does not converge in the reals.

there will not exist such a sequence in D that converges to 1 because x_n cannot equal 1 for any n. so the limit must exist. i mean why wouldn't the limit exist?
 
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  • #2
stripes said:

Homework Statement



Consider a function f: D∈R, where D = {1/n for natural numbers n (1, 2, 3, 4, etc.)} and f(x) = 3x - 1 for all x in D. Explain why the limit of f(x) as x → 1 does not exist.

Homework Equations





The Attempt at a Solution



Uh I figured it would exist. We know a function does not have a limit at c if and only if there exists a sequence (x_n) where x_n ≠ c for all natural numbers n such that (x_n) converges to c but the sequence (f(x_n)) does not converge in the reals.

there will not exist such a sequence in D that converges to 1 because x_n cannot equal 1 for any n. so the limit must exist.
x1 = 1/1 = 1, right?
stripes said:
i mean why wouldn't the limit exist?
 
  • #3
Mark44 said:
x1 = 1/1 = 1, right?

does this have to do with the fact that the limit from the left does not exist? but 1/n is always less than or equal to 1...so things seem well-defined...
 
  • #4
The limit from the left has a better chance of existing than the limit from the right, which is not to say that the limit from the left exists. It might be helpful to sketch a graph of f, keeping in mind that the inputs to f come from your set D.
 
  • #5
i don't quite understand the notion of a limit from one direction having a better chance of existing. to me, it is extremely clear the limit from the left does not exist, so the limit itself doesn't exist.

but things are confusing because we're using 1/n, which goes a different direction than x. if that makes sense.
 
  • #6
By "better chance" I wasn't implying that the limit from the left actually existed. What I was getting at is that there are numbers in D that are smaller than 1, but no numbers in D that are larger than 1, so there's no chance of a limit from the right existing.

stripes said:
but things are confusing because we're using 1/n, which goes a different direction than x. if that makes sense.
It might be helpful to look at set D like this:
D = {..., 1/5, 1/4, 1/3, 1/2, 1}
Note that you can never get closer to 1 than 1/2 for the numbers in D.
 

1. What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function as its input values approach a certain point. It represents the value that the function approaches, but may not necessarily reach, at a given point.

2. Why does the limit of a function not exist at x = 1?

When evaluating the limit of a function at a certain point, the function must approach the same value from both the left and right sides of the point. If the function approaches different values from the left and right sides, the limit does not exist. In this case, the limit of the function at x = 1 does not exist because the function approaches different values from the left and right sides of 1.

3. How do you explain the concept of one-sided limits?

A one-sided limit is the value that a function approaches from only one side of a given point. It is used when the function approaches different values from the left and right sides of the point, indicating that the overall limit does not exist. In this case, we can evaluate the one-sided limits to determine the behavior of the function near the point.

4. Can a function have a limit at a point but not be defined at that point?

Yes, it is possible for a function to have a limit at a point but not be defined at that point. This occurs when the function approaches a specific value as the input approaches the point, but the point is not in the domain of the function. In this case, the function is undefined at that point.

5. How can the graph of a function help explain why the limit does not exist at a certain point?

The graph of a function can visually show the behavior of the function as the input approaches a certain point. If the graph has a gap or hole at the given point, it indicates that the function is not continuous and the limit does not exist at that point. Similarly, if the graph has a vertical asymptote at the point, it also indicates that the limit does not exist as the function approaches infinity or negative infinity at that point.

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