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. Substitute the value into the numerator: 5−5*(5)+6=25−25+6=6

3. Substitute the value into the denominator: 5 - 5 = 0$.

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

5. Evaluate the one-sided limits to determine the behavior. As x→5 the expression 6/small positive approaches ∞ As x→5 the expression 6/small negative approaches −∞

6. Conclude that since the one-sided limits do not match and the expression grows 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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