Evaluate the Limit

Problem

(lim_x→−4)((1/4+1/x)/(4+x))

Solution

1. Identify the indeterminate form by substituting x=−4 into the expression, which results in 0/0

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

1/4+1/x=(x+4)/(4*x)

3. Substitute the simplified numerator back into the original limit expression.

(lim_x→−4)((x+4)/(4*x)/(4+x))

4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

(x+4)/(4*x)⋅1/(4+x)=1/(4*x)

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

1/(4*(−4))=−1/16

Final Answer

(lim_x→−4)((1/4+1/x)/(4+x))=−1/16

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