Evaluate the Limit

Problem

(lim_x→3)((1/x−1/3)/(x−3))

Solution

1. Identify the indeterminate form by substituting x=3 into the expression, which results in 0/0

2. Find a common denominator for the terms in the numerator to combine the fractions.

1/x−1/3=(3−x)/(3*x)

3. Substitute the simplified numerator back into the original limit expression.

(lim_x→3)((3−x)/(3*x)/(x−3))

4. Rewrite the complex fraction as a product by multiplying the numerator by the reciprocal of the denominator.

(lim_x→3)((3−x)/(3*x*(x−3)))

5. Factor out a negative sign from the numerator to reveal a common factor with the denominator.

3−x=−(x−3)

6. Cancel the common factor (x−3) from the numerator and the denominator, provided x≠3

(lim_x→3)((−(x−3))/(3*x*(x−3)))=(lim_x→3)((−1)/(3*x))

7. Evaluate the limit by substituting x=3 into the simplified expression.

(−1)/(3*(3))=−1/9

Final Answer

(lim_x→3)((1/x−1/3)/(x−3))=−1/9

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