lim (x⁻¹-1)⁄x-1 as x approaches 1.
I know the answer is -1 but I am not sure how to solve the numerator x⁻¹-1.
(x⁻¹-1) / x-1
x⁻¹-1 = 1/x - 1 = (1-x)/x
So, your fraction is just
(1-x)/x
---------- = -1/x
x-1
To solve the numerator of the expression (x⁻¹ - 1), you can simplify it by applying the rules of exponents.
Step 1: Rewrite x⁻¹ as 1/x. The exponent -1 is equivalent to taking the reciprocal of the base.
So, the expression becomes (1/x - 1).
Step 2: Find a common denominator for the fractions 1/x and 1.
The common denominator is x. So, we need to rewrite 1 as x/x.
Now, the expression becomes (1/x - x/x), which can be simplified as (1 - x)/x.
Now that you have simplified the numerator, the expression can be written as (1 - x)/x.
To find the limit of (1 - x)/x as x approaches 1, you can substitute 1 into the expression.
lim (x⁻¹ - 1)/x-1 as x approaches 1
= lim (1 - 1)/1
= lim 0/1
= 0.
Therefore, the limit of (x⁻¹ - 1)/x-1 as x approaches 1 is 0, not -1.
If you were told that the answer is -1, it is possible that there was a mistake in the statement or the solution provided. Double-check the problem statement and solution to ensure accuracy.