Evaluate the Limit

Problem

(lim_x→0)((x−3)/(x2−3*x))

Solution

1. Factor the denominator by taking out the greatest common factor, which is x

(x−3)/(x*(x−3))

2. Simplify the expression by canceling the common factor (x−3) from the numerator and the denominator, noting that x≠3

1/x

3. Evaluate the limit by observing the behavior of the simplified function as x approaches 0

(lim_x→0)(1/x)

4. Analyze the one-sided limits to determine if the general limit exists. As x approaches 0 from the right, the value goes to ∞ as x approaches 0 from the left, the value goes to −∞

(lim_x→0)(1/x)=∞

(lim_x→0)(1/x)=−∞

5. Conclude that because the one-sided limits are not equal, the limit does not exist.

Final Answer

(lim_x→0)((x−3)/(x2−3*x))=Does Not Exist

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