Evaluate Using L'Hospital's Rule

Problem

(lim_x→0)(sin(x))*ln(x)

Solution

1. Identify the indeterminate form. As x→0 sin(x)→0 and ln(x)→−∞ which results in the form 0⋅∞

2. Rewrite the expression as a fraction to prepare for L'Hospital's Rule. We move sin(x) to the denominator as csc(x)

(lim_x→0)(ln(x)/csc(x))

3. Verify the new form. Now the limit is in the form (−∞)/∞ which allows the use of L'Hospital's Rule.

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

(lim_x→0)(d(ln(x))/d(x)/d(csc(x))/d(x))

5. Compute the derivatives. The derivative of ln(x) is 1/x and the derivative of csc(x) is −csc(x)*cot(x)

(lim_x→0)(1/x/(−csc(x)*cot(x)))

6. Simplify the trigonometric expression using identities csc(x)=1/sin(x) and cot(x)=cos(x)/sin(x)

(lim_x→0)(1/x/(−cos(x)/sin2(x)))

7. Rearrange the fraction to make the limit easier to evaluate.

(lim_x→0)(−)sin2(x)/(x*cos(x))

8. Split the limit into known parts to evaluate.

(lim_x→0)(−)sin(x)/x⋅sin(x)/cos(x)

9. Evaluate the limit using the fundamental limit (lim_x→0)(sin(x)/x)=1

−1⋅0/1=0

Final Answer

(lim_x→0)(sin(x))*ln(x)=0

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