Evaluate the Limit ( limit as x approaches 2 of x-3)/(x-2)

Problem

(lim_x→2)((x−3)/(x−2))

Solution

1. Evaluate the numerator and denominator at the limit point x=2

2. Substitute x=2 into the numerator: 2 - 3 = -1$.

3. Substitute x=2 into the denominator: 2 - 2 = 0$.

4. Analyze the behavior of the fraction as x approaches 2 Since the numerator approaches a non-zero constant (−1 and the denominator approaches 0 the limit does not exist and tends toward infinity.

5. Check the one-sided limits to determine the direction. As x→2 the denominator is positive, so (−1)/0→−∞ As x→2 the denominator is negative, so (−1)/0→∞

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

Final Answer

(lim_x→2)((x−3)/(x−2))=Does Not Exist

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