Evaluate the Limit limit as x approaches 1 of (1/x-1)/(x-1)

Problem

(lim_x→1)((1/x−1)/(x−1))

Solution

1. Identify the form of the limit by substituting x=1 into the expression.

2. Substitute x=1 to find that the numerator is 1 - 1 = 0a*n*d(t)*h*e*d(e)*n*o*m*i*n*a*t*o*r*i*s() - 1 = 0,r*e*s(u)*l*t*i*n*g*i*n*t*h*e*i*n*d(e)*t*e*r*m*i*n*a*t*e*ƒ*o*r*mfrac{0}{0}$.

3. Simplify the numerator by finding a common denominator for the terms 1/x and 1

1/x−1=(1−x)/x

4. Rewrite the original limit expression using the simplified numerator.

(1−x)/x/(x−1)

5. Factor out a negative sign from the numerator to make it easier to cancel terms.

(1−x)/x=(−(x−1))/x

6. Divide the simplified numerator by the denominator (x−1)

(−(x−1))/(x*(x−1))

7. Cancel the common factor (x−1) from the numerator and the denominator, provided x≠1

(−1)/x

8. Evaluate the limit by substituting x=1 into the remaining expression.

(lim_x→1)((−1)/x)=−1

Final Answer

(lim_x→1)((1/x−1)/(x−1))=−1

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