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

Problem

(lim_x→3)(x/(x−3))

Solution

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

2. Substitute the value x=3 into the expression to check for a determinate form.

x→3

x−3→0

3. Determine the limit type. Since the numerator approaches 3 and the denominator approaches 0 the limit does not exist as a finite number and involves vertical asymptotes.

4. Evaluate the one-sided limits to determine the behavior from each direction.

5. Check the left-hand limit as x approaches 3 from the left (x<3. The numerator is positive and the denominator is negative.

(lim_x→3)(x/(x−3))=−∞

6. Check the right-hand limit as x approaches 3 from the right (x>3. The numerator is positive and the denominator is positive.

(lim_x→3)(x/(x−3))=∞

7. Conclude that since the one-sided limits do not match, the general limit does not exist.

Final Answer

(lim_x→3)(x/(x−3))=Does Not Exist

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