Evaluate the Limit

Problem

(lim_x→1)(x/ln(x))

Solution

1. Identify the form of the limit by substituting the value x=1 into the expression.

2. Evaluate the numerator as x approaches 1

(lim_x→1)(x)=1

3. Evaluate the denominator as x approaches 1

(lim_x→1)(ln(x))=0

4. Analyze the behavior of the fraction. Since the numerator approaches a non-zero constant (1 and the denominator approaches 0 the limit does not exist as a finite number.

5. Determine the one-sided limits. As x→1 ln(x) is positive and approaching 0 so the expression approaches +∞ As x→1 ln(x) is negative and approaching 0 so the expression approaches −∞

(lim_x→1)(x/ln(x))=∞

(lim_x→1)(x/ln(x))=−∞

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→1)(x/ln(x))=Does Not Exist

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