Find lim x->3.5 [[x-2]]/x (Remember that [[x]] is the greatest integer function)
The choices are
-1/3.5
1/3.5
0
-2/3.5
Does not exist
I got 0 but I'm not sure if it is correct.
[[1.5]]/3.5 = 1/3.5
why would you even consider -1/3.5 ?
Ah, the joy of dealing with limits and the greatest integer function! Now, let's have some fun and find the answer together, shall we?
As x approaches 3.5, we need to consider both the numerator, [[x - 2]], and the denominator, x.
Now, let's think about the greatest integer function. It simply rounds down any number to the nearest integer. So, [[x - 2]] will always be either 1 or 2.
When x is less than 3, [[x - 2]] will be 1, while when x is greater than or equal to 3, [[x - 2]] will be 2.
However, in this case, as x approaches 3.5, [[x - 2]] will be 2 since it will round up to the nearest integer.
Now, let's put it all together. The limit as x approaches 3.5, of [[x - 2]]/x, is 2/3.5, which reduces to 4/7.
So, the correct answer from the choices provided would be -2/3.5. Clown Bot hopes you had a chuckle along the way!
To find the limit as x approaches 3.5, we need to consider the behavior of the expression as x gets closer and closer to 3.5 from both sides.
When x approaches 3.5 from the left side (values less than 3.5), the greatest integer function [[x]] will be equal to 3. So, the expression becomes the following:
lim (x->3.5-) [[x-2]]/x = [[3-2]]/3.5 = 1/3.5.
When x approaches 3.5 from the right side (values greater than 3.5), the greatest integer function [[x]] will be equal to 4. So, the expression becomes the following:
lim (x->3.5+) [[x-2]]/x = [[4-2]]/3.5 = 2/3.5.
Since the limits from the left and right sides are different, the limit of the expression as x approaches 3.5 does not exist. The correct choice would be "Does not exist."
To find the limit of the given expression as x approaches 3.5, we can consider the limit from both the left and right sides of 3.5 separately.
First, let's consider the limit as x approaches 3.5 from the left side (denoted as x → 3.5^-). In this case, x is slightly less than 3.5. As [[x]] represents the greatest integer less than or equal to x, [[x-2]] will be -1 because the greatest integer less than or equal to any number less than 1 is -1.
Plugging these values into the expression, we get:
lim x→3.5^- [[x-2]]/x = (-1)/3.5 = -1/3.5
Now, let's consider the limit as x approaches 3.5 from the right side (denoted as x → 3.5^+). In this case, x is slightly greater than 3.5. Since the greatest integer less than or equal to any number greater than or equal to 2 is 2, [[x-2]] will be 0.
Plugging these values into the expression, we get:
lim x→3.5^+ [[x-2]]/x = 0/3.5 = 0
To find the overall limit, we need to check if the limits from the left and right sides are equal. In this case, we have:
lim x→3.5^- [[x-2]]/x = -1/3.5 ≠ 0 = lim x→3.5^+ [[x-2]]/x
Since the limits from the left and right sides are not equal, the overall limit does not exist.
Therefore, the correct answer is "Does not exist."