Find the limit, if it exists
lim x->0.1^- (10x-1)/(|10x^3-x^2|)
To find the limit as x approaches 0.1 from the left, we can substitute 0.1 into the expression and see if it approaches a finite value.
lim x->0.1^- (10x-1)/(|10x^3-x^2|)
Substituting x = 0.1:
lim x->0.1^- (10(0.1)-1)/(|10(0.1)^3-(0.1)^2|)
= lim x->0.1^- (1-1)/(0.001-0.01)
= lim x->0.1^- 0/-0.009
= 0/-0.009
Since the numerator is 0, and the denominator is negative, the limit is not defined. The limit does not exist.
To find the limit as x approaches 0.1 from the negative direction (x → 0.1^-), we need to evaluate the expression (10x - 1) / (|10x^3 - x^2|).
First, let's simplify the expression by factoring out x from the numerator and the denominator:
(10x - 1) / (|10x^3 - x^2|) = x(10 - 1/x) / (|x^2(10x - 1)|)
Now, we can analyze each factor separately. Let's start with the factor in the numerator: x(10 - 1/x).
As x approaches 0.1 from the negative direction, x becomes close to 0. Therefore, (10 - 1/x) becomes 10.
Now, let's analyze the factor in the denominator: |x^2(10x - 1)|.
As x approaches 0.1 from the negative direction, x becomes close to 0. Thus, (10x - 1) also becomes close to -1. Therefore, the absolute value of this expression is equal to |-1| = 1.
Now, we have:
x(10 - 1/x) / (|x^2(10x - 1)|) = x(10) / (1) = 10x
Therefore, the limit as x approaches 0.1 from the negative direction of (10x - 1) / (|10x^3 - x^2|) is 10(0.1) = 1.
To find the limit of a function as x approaches a certain value, you can apply the properties of limits and algebraic techniques. In this case, let's find the limit as x approaches 0.1 from the left.
First, substitute the value into the function:
lim x->0.1^- (10x-1)/(|10x^3-x^2|)
As x approaches 0.1 from the left, you need to determine the value of the expression.
Next, simplify the expression by factoring and canceling out common factors:
lim x->0.1^- (10x-1)/(|x^2|(10x-1))
Since the denominator has an absolute value, we can rewrite the expression as:
lim x->0.1^- (10x-1)/(x^2 * |10x-1|)
Now, we can cancel out the common factor of (10x-1) from the numerator and denominator:
lim x->0.1^- 1/(x^2)
Finally, evaluate the limit as x approaches 0.1 from the left:
lim x->0.1^- 1/(0.1^2)
lim x->0.1^- 1/0.01
lim x->0.1^- 100
Therefore, the limit as x approaches 0.1 from the left is 100.