Evaluate the Limit

Problem

(lim_x→1)(x/(x−1)−1/ln(x))

Solution

1. Combine the fractions into a single expression by finding a common denominator.

(x*ln(x)−(x−1))/((x−1)*ln(x))

2. Evaluate the limit at x=1 to check for an indeterminate form.

(1*ln(1)−(1−1))/((1−1)*ln(1))=0/0

3. Apply L'Hôpital's Rule by differentiating the numerator and the denominator with respect to x

(lim_x→1)(d(x*ln(x)−x+1)/d(x)/d((x−1)*ln(x))/d(x))

4. Differentiate the expressions using the product rule.

(lim_x→1)((ln(x)+x⋅1/x−1)/(ln(x)+(x−1)⋅1/x))

5. Simplify the resulting expression.

(lim_x→1)(ln(x)/(ln(x)+1−1/x))

6. Evaluate the limit again at x=1

ln(1)/(ln(1)+1−1/1)=0/0

7. Apply L'Hôpital's Rule a second time.

(lim_x→1)(d(ln(x))/d(x)/d(ln(x)+1−x(−1))/d(x))

8. Differentiate the numerator and denominator.

(lim_x→1)(1/x/(1/x+1/(x2)))

9. Substitute x=1 into the simplified derivative expression.

1/1/(1/1+1/1)=1/2

Final Answer

(lim_x→1)(x/(x−1)−1/ln(x))=1/2

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