Evaluate the Limit

Problem

(lim_x→0)((1/(2+x)−1/2)/x)

Solution

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

2. Find a common denominator for the terms in the numerator to combine the fractions.

1/(2+x)−1/2=(2−(2+x))/(2*(2+x))

3. Simplify the numerator of the complex fraction by distributing the negative sign and combining like terms.

(2−2−x)/(2*(2+x))=(−x)/(2*(2+x))

4. Rewrite the entire limit expression by placing the simplified numerator over the original denominator x

(lim_x→0)((−x)/(2*(2+x))/x)

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

(lim_x→0)((−x)/(2*(2+x)))⋅1/x

6. Cancel the common factor of x from the numerator and the denominator, provided x≠0

(lim_x→0)((−1)/(2*(2+x)))

7. Evaluate the limit by substituting x=0 into the simplified expression.

(−1)/(2*(2+0))=−1/4

Final Answer

(lim_x→0)((1/(2+x)−1/2)/x)=−1/4

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