Evaluate Using L'Hospital's Rule

Problem

(lim_x→0)(x)*ln(x)

Solution

1. Identify the indeterminate form. As x→0 the term x approaches 0 and ln(x) approaches −∞ resulting in the indeterminate form 0⋅(−∞)

2. Rewrite the expression as a fraction to prepare for L'Hospital's Rule. Move x to the denominator as 1/x to create the form (−∞)/∞

(lim_x→0)(ln(x)/1/x)

3. Apply L'Hospital's Rule by differentiating the numerator and the denominator separately.

d(ln(x))/d(x)=1/x

d()/d(x)1/x=−1/(x2)

4. Substitute the derivatives back into the limit.

(lim_x→0)(1/x/(−1/(x2)))

5. Simplify the complex fraction by multiplying by the reciprocal.

1/x⋅(−x2)=−x

6. Evaluate the limit as x approaches 0

(lim_x→0)(−)*x=0

Final Answer

(lim_x→0)(x)*ln(x)=0

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