Evaluate the Limit

Problem

(lim_x→4)((x−4)/(x2−3*x−4))

Solution

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

(4−4)/(4−3*(4)−4)=0/(16−12−4)=0/0

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

x2−3*x−4=(x−4)*(x+1)

3. Rewrite the limit using the factored form of the denominator.

(lim_x→4)((x−4)/((x−4)*(x+1)))

4. Cancel the common factor of (x−4) from the numerator and the denominator, noting that x≠4 as it approaches the limit.

(lim_x→4)(1/(x+1))

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

1/(4+1)=1/5

Final Answer

(lim_x→4)((x−4)/(x2−3*x−4))=1/5

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