Evaluate the Limit

Problem

(lim_x→6)((5*x2−29*x−6)/(x2+19*x−150))

Solution

1. Substitute the value x=6 into the expression to check for an indeterminate form.

(5*(6)2−29*(6)−6)/((6)2+19*(6)−150)=(180−174−6)/(36+114−150)=0/0

2. Factor the numerator 5*x2−29*x−6 by finding two numbers that multiply to −30 and add to −29 which are −30 and 1

5*x2−30*x+x−6=5*x*(x−6)+1*(x−6)=(5*x+1)*(x−6)

3. Factor the denominator x2+19*x−150 by finding two numbers that multiply to −150 and add to 19 which are 25 and −6

x2+19*x−150=(x+25)*(x−6)

4. Simplify the limit expression by canceling the common factor (x−6) from the numerator and the denominator.

(lim_x→6)(((5*x+1)*(x−6))/((x+25)*(x−6)))=(lim_x→6)((5*x+1)/(x+25))

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

(5*(6)+1)/(6+25)=31/31=1

Final Answer

(lim_x→6)((5*x2−29*x−6)/(x2+19*x−150))=1

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