Evaluate Using L'Hospital's Rule limit as x approaches 0 of x^(sin(x))

Problem

(lim_x→0)(xsin(x))

Solution

1. Identify the indeterminate form. As x→0 the expression xsin(x) takes the form 0

2. Rewrite the expression using the natural logarithm and exponential function to prepare for L'Hospital's Rule.

y=xsin(x)

ln(y)=ln(xsin(x))

ln(y)=sin(x)*ln(x)

3. Transform the limit into a quotient form ∞/∞ to apply L'Hospital's Rule.

(lim_x→0)(ln(y))=(lim_x→0)(ln(x)/csc(x))

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

(lim_x→0)(d(ln(x))/d(x)/d(csc(x))/d(x))=(lim_x→0)(1/x/(−csc(x)*cot(x)))

5. Simplify the resulting trigonometric expression.

(lim_x→0)(1/(−x⋅1/sin(x)⋅cos(x)/sin(x)))=(lim_x→0)(−)sin2(x)/(x*cos(x))

6. Evaluate the limit by splitting the terms.

(lim_x→0)(−)sin(x)/x⋅sin(x)/cos(x)

(lim_x→0)(sin(x)/x)=1

(lim_x→0)(sin(0)/cos(0))=0/1=0

(lim_x→0)(ln(y))=−1⋅0=0

7. Solve for the original limit by exponentiating the result.

(lim_x→0)(y)=e0=1

Final Answer

(lim_x→0)(xsin(x))=1

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