Find the indicated limits.
Lim x-> 1- sqrt 1 - 1/ x-1
The solution cannot be deciphered without some grouping symbols to indicate what the problem is.
Lim x approaching 1-
(Square root 1-1)/( x-1)
it still is not clear. (sqrt 1 -1) is zero.
Yes, I know. The book has the answer as 1/2. Maybe book is wrong.
To find the limit of the expression as x approaches 1 from the left (denoted as x → 1-), we will substitute the value of x into the expression and simplify.
Lim x → 1- sqrt(1 - 1/(x-1))
First, observe that the expression inside the square root, 1 - 1/(x-1), can be simplified further. We can find a common denominator for 1 and 1/(x-1):
1 - 1/(x-1) = (x-1)/(x-1) - 1/(x-1)
= (x-1 - 1)/(x-1)
= (x-2)/(x-1)
Now, substitute this simplified expression back into the original limit:
Lim x → 1- sqrt((x-2)/(x-1))
To evaluate the limit, we can substitute the value x = 1 into the expression. However, observe that when x → 1-, the term (x-2) approaches -1 and the term (x-1) approaches 0. Therefore, we are left with an expression of the form sqrt(-1/0), which is undefined.
Hence, the limit as x approaches 1 from the left of sqrt(1 - 1/(x-1)) is undefined.