I have no clue how to do this problem:
lim---->2 then thees this little dash slightly above the 2 and then its |x-2|/(x-2)
so you want the left limit (as x->2 from below)
Now, recall the definition of |u|
|u| = u if u >= 0
|u| = -u if u < 0
When x < 2, x-2 < 0, so |x-2| = 2-x
So, the limit is (2-x)/(x-2) -> -1
You can see this on the graph at
http://www.wolframalpha.com/input/?i=%7Cx-2%7C%2F(x-2)
THANK YOU!!!!!!!!!!!
The problem you have described is asking you to find the limit as x approaches 2 of the function |x-2|/(x-2). Let's break it down step by step:
1. Start by plugging in the value 2 into the function. This will help us determine if the expression is defined at x = 2. In this case, both the numerator and denominator would be 0, which means we have an indeterminate form.
2. Next, we need to simplify the expression so that we can evaluate the limit. Since the numerator |x-2| represents the absolute value of x-2, it will always be non-negative. Therefore, we can rewrite |x-2| as (x-2) if x is greater than or equal to 2, and as -(x-2) if x is less than 2.
3. Now, rewrite the function with the simplified numerator and denominator. We have (x-2)/(x-2) if x is greater than or equal to 2, and -(x-2)/(x-2) if x is less than 2.
4. Simplify further by canceling out the common factors in the numerator and denominator. The function becomes 1 if x is greater than or equal to 2, and -1 if x is less than 2.
5. Finally, evaluate the limit by considering the behavior of the function as x approaches 2 from both sides. From the left side (x < 2), the function approaches -1, and from the right side (x > 2), the function approaches 1. Therefore, the limit as x approaches 2 of |x-2|/(x-2) does not exist because the limit values from the left and right sides are not the same.
In summary, the function |x-2|/(x-2) does not have a limit as x approaches 2.