Use the graph of the function f of x equals 2 plus 1 over x to determine which of the following statements is false for the sequence a sub n equals the sequence 2 plus 1 over n . (2 points)
The sequence is monotonic and bounded.
The sequence is bounded but not convergent.
The sequence is unbounded but convergent.
The sequence is unbounded.
Please Explain
so look at the graph
is it always increasing or decreasing?
does it have a horizontal asymptote? (That would make it bounded)
To determine which of the statements is false for the sequence a sub n = 2 + 1/n, we can analyze the behavior of the function f(x) = 2 + 1/x and the graph of this function.
The graph of the function f(x) = 2 + 1/x is a hyperbola with a vertical asymptote at x=0. As x approaches 0 from the right (x > 0), f(x) approaches positive infinity, and as x approaches 0 from the left (x < 0), f(x) approaches negative infinity.
Now let's compare this to the sequence a sub n = 2 + 1/n. As n approaches infinity, 1/n approaches 0. Therefore, a sub n approaches 2 + 0 = 2 as n approaches infinity. This means that the sequence is convergent and its limit is 2.
Now let's analyze the statements:
1. The sequence is monotonic and bounded.
This statement is true. Since the sequence converges to 2 as n approaches infinity, it is also monotonic and bounded.
2. The sequence is bounded but not convergent.
This statement is false. The sequence is both bounded and convergent, as explained above.
3. The sequence is unbounded but convergent.
This statement is false. The sequence is bounded, not unbounded.
4. The sequence is unbounded.
This statement is false. The sequence is bounded, not unbounded.
Therefore, the false statement is "The sequence is unbounded."
To determine which statement is false for the sequence a sub n equals 2 plus 1 over n, we need to analyze the behavior of the function f(x) = 2 + 1/x and how it relates to the given sequence.
First, let's consider the graph of the function f(x) = 2 + 1/x. The graph will be a hyperbola that opens upwards in the first and third quadrants. The function approaches the x-axis but never touches it. As x approaches positive infinity, the function approaches y = 2 (the horizontal asymptote). Similarly, as x approaches negative infinity, the function approaches y = 2.
Now, let's examine each statement:
1. The sequence is monotonic and bounded.
To determine if the sequence is monotonic, we need to check if it is either always increasing or always decreasing. Looking at a sub n = 2 + 1/n, as n increases, the value of the sequence decreases. Therefore, it is not monotonic.
Next, let's investigate if the sequence is bounded. From the graph of f(x), we can observe that as x gets larger, the value of f(x) approaches 2, and similarly, as x decreases, f(x) approaches 2. This indicates that the sequence a sub n = 2 + 1/n is bounded between 2 and 3 (excluding 2). Therefore, the sequence is bounded.
2. The sequence is bounded but not convergent.
From the analysis above, we concluded that the sequence is bounded. However, the sequence is not convergent because it does not approach a specific number as n goes to infinity. Instead, the sequence fluctuates between the values of 2 and 3 (excluding 2).
3. The sequence is unbounded but convergent.
This statement contradicts what we have deduced from the graph and the given sequence. The sequence is bounded but not convergent. Therefore, this statement is false.
4. The sequence is unbounded.
Again, referring to the graph of f(x) and the sequence, we can see that as n approaches infinity, the values of the sequence approach 2, which means it is bounded between 2 and 3 (excluding 2). Therefore, this statement is also false.
In conclusion, the false statement for the sequence a sub n = 2 + 1/n is "The sequence is unbounded but convergent."