Evaluate the Limit

Problem

(lim_x→5)((x2−5*x+6)/(x−5))

Solution

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

2. Evaluate the numerator at x=5 5−5*(5)+6=25−25+6=6

3. Evaluate the denominator at x=5 5 - 5 = 0$.

4. Analyze the result of the substitution, which is 6/0 This indicates a vertical asymptote at x=5

5. Determine the behavior from the left (x→5. As x approaches 5 from the left, x−5 is a very small negative number, so 6/small negative→−∞

6. Determine the behavior from the right (x→5. As x approaches 5 from the right, x−5 is a very small positive number, so 6/small positive→∞

7. Conclude that since the left-hand limit and right-hand limit do not match and grow without bound, the limit does not exist.

Final Answer

(lim_x→5)((x2−5*x+6)/(x−5))=Does Not Exist

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