lim
X--> 0 from the left
(9/x - 9/|x|)
Find the limit if it exists
Well, since |x| = -x when x<0, you have 9/x -9/(-x) = 18/x = -∞
Well, isn't this a tricky mathematical circus act! Let's see if we can untangle the clown-like complexity of this problem.
To evaluate the limit as x approaches 0 from the left of the expression (9/x - 9/|x|), we need to consider two cases:
Case 1: x > 0
In this case, the expression becomes (9/x - 9/x) = 0.
Case 2: x < 0
When x is negative, we have to be cautious with the absolute value |x|. Since x is approaching 0 from the left, |x| will be equal to -x. So the expression becomes (9/x - 9/(-x)) = (9/x + 9/x) = 18/x.
Now, as x approaches 0 from the left, we can clearly see that in both cases, the expression approaches different values. Case 1 approaches 0, while Case 2 approaches infinity (since 18 divided by a very small number is quite large).
Since the limit does not converge to a single value, we conclude that the limit does not exist. And just like a clown juggling balls, this problem keeps us on our toes!
To find the limit of the expression (9/x - 9/|x|) as X approaches 0 from the left, we need to determine the behavior of the expression as X gets closer and closer to 0 from values less than 0.
First, let's simplify the expression:
9/x - 9/|x|
Now, consider the absolute value function |x|. When X is negative, the absolute value of X is equal to -X. Therefore, |x| = -x when X < 0.
Now, substitute -x for |x| in the expression:
9/x - 9/(-x)
Simplify further:
9/x + 9/x
Combine the fractions with a common denominator:
(9 + 9)/x
Simplify:
18/x
Now, as X approaches 0 from the left, the value of X becomes negative and very close to 0. Therefore, the expression (18/x) approaches negative infinity.
Hence, the limit of (9/x - 9/|x|) as X approaches 0 from the left, if it exists, is negative infinity.
To find the limit of the expression (9/x - 9/|x|) as x approaches 0 from the left, we can follow these steps:
Step 1: Simplify the expression.
The expression (9/x - 9/|x|) can be simplified by expressing it as a single fraction with a common denominator. The common denominator can be chosen as |x|. To do this, multiply the first term by |x|/|x|:
= (9|x|/|x|^2 - 9/|x|)
Step 2: Combine the fractions.
Now that we have a common denominator, we can combine the fractions:
= (9|x| - 9|x|^2)/|x|^2
Step 3: Factor out the common multiples.
Notice that in the numerator, we can factor out a common multiple of |x|:
= 9|x|(1 - |x|)/|x|^2
Step 4: Simplify further.
Simplify the expression by canceling out the common factors:
= 9(1 - |x|)/|x|
Step 5: Take the limit.
To find the limit as x approaches 0 from the left, we substitute x = 0 into the simplified expression:
= 9(1 - 0)/0
Since we have a denominator of 0, the expression becomes undefined. Therefore, the limit does not exist.