Evaluate the Limit limit as x approaches 3 of (3x-5)/(x-3)

Problem

(lim_x→3)((3*x−5)/(x−3))

Solution

1. Analyze the behavior of the numerator as x approaches 3

(lim_x→3)(3*x−5)=3*(3)−5=4

2. Analyze the behavior of the denominator as x approaches 3

(lim_x→3)(x−3)=3−3=0

3. Determine the type of limit. Since the numerator approaches a non-zero constant (4 and the denominator approaches 0 the limit does not exist and will involve vertical asymptotes.

4. Evaluate the one-sided limit from the left (x→3. As x approaches 3 from values slightly less than 3 the numerator is positive (≈4 and the denominator is negative (e.g., 2.9 - 3 = -0.1$).

(lim_x→3)((3*x−5)/(x−3))=−∞

5. Evaluate the one-sided limit from the right (x→3. As x approaches 3 from values slightly greater than 3 the numerator is positive (≈4 and the denominator is positive (e.g., 3.1 - 3 = 0.1$).

(lim_x→3)((3*x−5)/(x−3))=∞

6. Conclude that because the left-hand and right-hand limits are not equal, the two-sided limit does not exist.

Final Answer

(lim_x→3)((3*x−5)/(x−3))=Does Not Exist

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