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. Simplify the numerator by finding a common denominator for the terms 1/4 and 1/x

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

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

(lim_x→−4)((x+4)/(4*x)/(4+x))

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

(x+4)/(4*x)⋅1/(4+x)=(x+4)/(4*x*(4+x))

5. Cancel the common factor (x+4) from the numerator and the denominator, noting that x≠−4 during the limit process.

1/(4*x)

6. 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

Want more problems? Check here!