Evaluate the Limit

Problem

(lim_x→0)((5*(3*x+6))/(5*(x+6)*(−6*x2−20*x−16)))

Solution

1. Simplify the expression by canceling the constant factor 5 from the numerator and the denominator.

(5*(3*x+6))/(5*(x+6)*(−6*x2−20*x−16))=(3*x+6)/((x+6)*(−6*x2−20*x−16))

2. Evaluate the limit by substituting x=0 into the simplified expression, as the function is continuous at this point.

(3*(0)+6)/((0+6)*(−6*(0)2−20*(0)−16))

3. Simplify the numerator and the denominator separately.

6/((6)*(−16))

4. Reduce the fraction by canceling the common factor 6

1/(−16)

Final Answer

(lim_x→0)((5*(3*x+6))/(5*(x+6)*(−6*x2−20*x−16)))=−1/16

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