find the limit(if it exists), if not explain why.
lim
x--> 4- (3[[x]]-5)
can't make out your limit expression
limit as x-->4- (from the left) of
(3[[x]]-5)
[[absolute value.]]
In any limit, one of the first things you should do is simply substitute.
If you get a real answer, that is your answer, you are done!
so
Lime 3|x| - 5 as x ---> 4 -
= 3(4)-5 = 7
Try a value near 4 from the left
e.g. x = 3.9999
then 3|3.9999|-5 = 6.9997 , yup, close enough for 7
To find the limit of a function as x approaches a certain value, you need to evaluate the function at values very close to that value from both sides, the left side and the right side. In this case, we want to find the limit as x approaches 4 from the left side.
The expression you provided is lim(x->4-) (3[[x]]-5), where [[x]] denotes the greatest integer less than or equal to x.
To evaluate this limit, we need to determine the value of the expression as x approaches 4 from the left side. Since we are considering the left side, x must be less than 4.
For example, let's substitute x = 3.9 into the expression:
3[[3.9]]-5 = 3*3 - 5 = 4
Now, let's substitute x = 3.5 into the expression:
3[[3.5]]-5 = 3*3 - 5 = 4
Similarly, for any value of x that is slightly less than 4, the value of the expression will be 4.
Therefore, the limit as x approaches 4 from the left side is 4.
Note: It is important to note that the limit does not depend on the actual value of f(4). It solely depends on the behavior of the function as x approaches 4 from the left side.