Evaluate the Limit

Problem

(lim_x→0)(x2)*ln(x)

Solution

1. Identify the indeterminate form. As x approaches 0 from the right, x2 approaches 0 and ln(x) approaches −∞ resulting in the form 0⋅(−∞)

2. Rewrite the expression to create a fraction suitable for L'Hôpital's Rule. Move x2 to the denominator as x(−2)

(lim_x→0)(ln(x)/(x(−2)))

3. Verify the new indeterminate form. The expression is now in the form (−∞)/∞

4. Apply L'Hôpital's Rule by differentiating the numerator and the denominator separately.

(lim_x→0)(d(ln(x))/d(x)/d(x(−2))/d(x))

5. Calculate the derivatives.

(lim_x→0)(1/x/(−2*x(−3)))

6. Simplify the resulting algebraic expression.

(lim_x→0)(1/x)⋅(x3)/(−2)

(lim_x→0)(−)(x2)/2

7. Evaluate the limit by substituting x=0

−0/2=0

Final Answer

(lim_x→0)(x2)*ln(x)=0

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