Evaluate the Limit limit as x approaches 5 of (x+1)/(x-5)

Problem

(lim_x→5)((x+1)/(x−5))

Solution

1. Identify the type of limit by substituting the value x=5 into the expression.

2. Evaluate the numerator as x approaches 5 which results in 5 + 1 = 6$.

3. Evaluate the denominator as x approaches 5 which results in 5 - 5 = 0$.

4. Analyze the behavior of the fraction 6/0 which indicates a vertical asymptote at x=5

5. Check the one-sided limits to determine if the limit exists.

6. Observe that as x→5 the denominator is positive and small, so (x+1)/(x−5)→∞

7. Observe that as x→5 the denominator is negative and small, so (x+1)/(x−5)→−∞

8. Conclude that since the one-sided limits do not match and are infinite, the limit does not exist.

Final Answer

(lim_x→5)((x+1)/(x−5))=Does Not Exist

Want more problems? Check here!