(3x |x-2|) / x-2
What is the limit as x approaches 2 from the left side?
while x < 2, -2 < 0, so, |x-2| = -(x-2)
So, you have the limit of
(3x(-(x-2)))/(x-2) = -3x
You can cancel the x-2 since x is not actually 2.
The limit is thus -6
The limit from the right, of course would be +6
See the graph at
http://www.wolframalpha.com/input/?i=%283x+|x-2|%29+%2F+%28x-2%29
To find the limit as x approaches 2 from the left side of the given expression (3x |x-2|) / (x-2), follow these steps:
Step 1: Substitute the value of x = 2 into the expression:
(3(2) |2-2|) / (2-2)
Step 2: Simplify the expression:
(6 |0|) / 0
Step 3: Determine the value within the absolute value sign (|0|):
The absolute value of 0, |0|, is equal to 0.
Step 4: Simplify the expression further:
(6 * 0) / 0
Step 5: Evaluate the expression:
0 / 0 is an indeterminate form, which means that the expression is undefined.
Therefore, the limit as x approaches 2 from the left side of the given expression is undefined.