Evaluate the Limit

Problem

(lim_x→−1)((x2−4*x)/(x2−3*x−4))

Solution

1. Evaluate the limit by direct substitution to check for an indeterminate form.

((−1)2−4*(−1))/((−1)2−3*(−1)−4)

(1+4)/(1+3−4)

5/0

2. Analyze the result of the substitution. Since the numerator is a non-zero constant (5 and the denominator is 0 the limit does not exist as a finite number and involves a vertical asymptote at x=−1

3. Factor the denominator to determine the behavior of the function near the asymptote.

x2−3*x−4=(x−4)*(x+1)

4. Examine the one-sided limits as x approaches −1

(lim_x→−1)((x*(x−4))/((x−4)*(x+1)))=(lim_x→−1)(x/(x+1))=(−1)/0=∞

(lim_x→−1)((x*(x−4))/((x−4)*(x+1)))=(lim_x→−1)(x/(x+1))=(−1)/0=−∞

5. Conclude that because the left-hand and right-hand limits do not match, the two-sided limit does not exist.

Final Answer

(lim_x→−1)((x2−4*x)/(x2−3*x−4))=Does Not Exist

Want more problems? Check here!