Evaluate the Limit

Problem

(lim_x→1)((2−x)/((x−1)2))

Solution

1. Analyze the behavior of the numerator as x approaches 1

(lim_x→1)(2−x)=2−1=1

2. Analyze the behavior of the denominator as x approaches 1

(lim_x→1)(x−1)=(1−1)2=0

3. Determine the sign of the expression near the limit point. Since the numerator approaches 1 (a positive value) and the denominator (x−1)2 is always positive for all x≠1 the fraction will grow without bound in the positive direction.

4. Conclude that as the denominator approaches 0 from the positive side while the numerator remains positive, the limit is positive infinity.

Final Answer

(lim_x→1)((2−x)/((x−1)2))=∞

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