Evaluating the Limit of (x^2 - |x-1|) / (x-1) as x Approaches 1 from the Left

To evaluate the limit of the given expression, we can simplify it using algebraic manipulations.

First, let's rewrite the expression:

lim(x→1-) (x^2 - |x-1|) / (x-1)

Next, we can simplify the numerator by considering two cases: when x < 1 and when x > 1.

When x < 1, |x-1| = -(x-1) = 1-x. So the numerator becomes:

(x^2 - |x-1|) = (x^2 - (1-x)) = (x^2 + x - 1)

When x > 1, |x-1| = x-1. So the numerator becomes:

(x^2 - |x-1|) = (x^2 - (x-1)) = (x^2 - x + 1)

Now, let's rewrite the limit using the simplified numerator:

lim(x→1-) [(x^2 + x - 1) / (x-1)] = lim(x→1-) [(x+1)(x-1) / (x-1)]

Notice that (x-1) cancels out in the numerator and denominator. Now, we can evaluate the limit:

lim(x→1-) [(x+1)(x-1) / (x-1)] = lim(x→1-) (x+1) = 2

Therefore, the limit of the given expression as x approaches 1 from the left (x→1-) is equal to 2.