Evaluate the Limit limit as x approaches 2 of (x-2)/(x^2-4)

Problem

(lim_x→2)((x−2)/(x2−4))

Solution

1. Identify the form of the limit by substituting x=2 into the expression, which results in the indeterminate form 0/0

2. Factor the denominator using the difference of squares formula, a2−b2=(a−b)*(a+b)

x2−4=(x−2)*(x+2)

3. Simplify the expression by canceling the common factor (x−2) from the numerator and the denominator.

(x−2)/((x−2)*(x+2))=1/(x+2)

4. Evaluate the limit by substituting x=2 into the simplified expression.

(lim_x→2)(1/(x+2))=1/(2+2)

5. Calculate the final numerical value.

1/4

Final Answer

(lim_x→2)((x−2)/(x2−4))=1/4

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