Evaluate the Limit limit as x approaches 2 of 1/((x-2)^2)

Problem

(lim_x→2)(1/((x−2)2))

Solution

1. Identify the form of the limit by substituting the value x=2 into the expression.

2. Observe that as x approaches 2 the denominator (x−2)2 approaches 0

3. Determine the sign of the expression by noting that (x−2)2 is always positive for any x≠2 because any non-zero real number squared is positive.

4. Conclude that since the numerator is a positive constant 1 and the denominator is a positive value approaching 0 the fraction grows without bound.

Final Answer

(lim_x→2)(1/((x−2)2))=∞

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