Evaluate the Limit

Problem

(lim_x→2)(sin(x−2)/(x2−4))

Solution

1. Identify the indeterminate form by substituting x=2 into the expression, which results in sin(0)/(4−4)=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 limit expression to isolate the standard limit form sin(θ)/θ

(lim_x→2)(sin(x−2)/(x−2)⋅1/(x+2))

4. Apply the limit product rule and the fundamental trigonometric limit (lim_u→0)(sin(u)/u)=1 where u=x−2

(lim_x→2)(sin(x−2)/(x−2))=1

5. Evaluate the remaining part of the expression by substituting x=2

(lim_x→2)(1/(x+2))=1/(2+2)=1/4

6. Multiply the results of the two limits to find the final value.

1⋅1/4=1/4

Final Answer

(lim_x→2)(sin(x−2)/(x2−4))=1/4

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