compute: limit as x approaches 0 (|4x-1|-|4x+1|)/x)
To compute the limit as x approaches 0 of the expression |4x-1|-|4x+1| divided by x, we’ll consider the left and right-hand limits separately
To compute the limit as x approaches 0 of the expression |4x-1|-|4x+1| divided by x, we’ll consider the left and right-hand limits separately.
First, let’s focus on the left-hand limit (as x approaches 0 from the negative side, denoted as x→0-). In this case, x<0. Thus, |x| = -x for x<0. We can rewrite the expression as:
(|4x-1| - |4x+1|)/x = ((4x-1) - (-(4x+1)))/x = (4x-1 + 4x + 1)/x = (8x)/x = 8
Now, let's look at the right-hand limit (as x approaches 0 from the positive side, denoted as x→0+). In this case, x>0. So, |x| = x for x>0. We can rewrite the expression as:
(|4x-1| – |4x+1|)/x = ((4x-1) – (4x+1))/x = (4x-1 – 4x – 1)/x = (-2)/x
Now, to find the limit, we need to consider what happens to these expressions as x approaches 0.
For the left-hand limit, as x approaches 0, the expression becomes 8. So, the left-hand limit is 8.
For the right-hand limit, as x approaches 0, the expression becomes (-2)/0, which is undefined since we cannot divide by zero.
Since the left-hand limit and the right-hand limit do not agree, the limit of the given expression as x approaches 0 does not exist.
In summary, the limit as x approaches 0 of (|4x-1|-|4x+1|)/x is undefined.
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