Evaluate the Limit

Problem

(lim_x→−2)((1/2+1/x)/(2+x))

Solution

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

2. Simplify the numerator by finding a common denominator for 1/2 and 1/x which is 2*x

1/2+1/x=(x+2)/(2*x)

3. Rewrite the original limit expression using the simplified numerator.

(lim_x→−2)((x+2)/(2*x)/(2+x))

4. Divide the fractions by multiplying the numerator by the reciprocal of the denominator.

(x+2)/(2*x)⋅1/(x+2)

5. Cancel the common factor of x+2 from the numerator and the denominator, provided x≠−2

1/(2*x)

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

1/(2*(−2))=−1/4

Final Answer

(lim_x→−2)((1/2+1/x)/(2+x))=−1/4

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