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

Problem

(lim_x→2)((2−x)/(x2−4))

Solution

1. Identify the indeterminate form by substituting x=2 into the expression, which results in 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. Rewrite the numerator to match a factor in the denominator by factoring out a negative sign.

2−x=−(x−2)

4. Substitute these expressions back into the limit.

(lim_x→2)((−(x−2))/((x−2)*(x+2)))

5. Simplify the fraction by canceling the common factor (x−2) provided x≠2

(lim_x→2)((−1)/(x+2))

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

(−1)/(2+2)=−1/4

Final Answer

(lim_x→2)((2−x)/(x2−4))=−1/4

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