Evaluate the Limit

Problem

(lim_x→4)((x2−4*x)/(x2−3*x−4))

Solution

1. Substitute the value x=4 into the expression to check for an indeterminate form.

(4−4*(4))/(4−3*(4)−4)=(16−16)/(16−12−4)=0/0

2. Factor the numerator by taking out the greatest common factor x

x2−4*x=x*(x−4)

3. Factor the quadratic trinomial in the denominator by finding two numbers that multiply to −4 and add to −3

x2−3*x−4=(x−4)*(x+1)

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

(x*(x−4))/((x−4)*(x+1))=x/(x+1)

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

4/(4+1)=4/5

Final Answer

(lim_x→4)((x2−4*x)/(x2−3*x−4))=4/5

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