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)

2. Simplify the numerator and denominator.

(1+4)/(1+3−4)=5/0

3. Analyze the result of the substitution. Since the numerator is a non-zero constant and the denominator is zero, the limit does not exist as a finite number and represents a vertical asymptote.

4. Check the behavior from the left (x→−1. As x approaches −1 from the left, x2−4*x is positive (≈5 and x2−3*x−4 is positive, so the limit approaches ∞

(lim_x→−1)((x2−4*x)/((x−4)*(x+1)))=∞

5. Check the behavior from the right (x→−1. As x approaches −1 from the right, x2−4*x is positive (≈5 and x2−3*x−4 is negative, so the limit approaches −∞

(lim_x→−1)((x2−4*x)/((x−4)*(x+1)))=−∞

6. Conclude that because the one-sided limits do not match, the general limit does not exist.

Final Answer

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

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