lim_(theta rarr0^(+))(sin(pi-theta))^(theta)

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Recognize symmetry of sine function: To solve the limit lim_(θ→0^+)(sin(π - θ))^θ, we first need to recognize that sin(π - θ) is equivalent to sin(θ) due to the symmetry of the sine function.Rewrite expression with limit: Since sin(θ) is continuous and differentiable at θ = 0, and sin(0) = 0, we can rewrite the expression as (sin(θ))^θ.Apply exponential and logarithmic functions: We can now apply the limit to the expression. However, the direct substitution of θ = 0 would result in the indeterminate form 0^0. To resolve this, we can use the exponential and logarithmic functions to transform the expression.Take natural logarithm of expression: Let's set y = (sin(θ))^θ and take the natural logarithm of both sides to get ln(y) = θln(sin(θ)).Apply L'Hôpital's Rule: Now we can find the limit of ln(y) as θ approaches 0 from the positive side: lim_(θ→0^+)θln(sin(θ)).Evaluate limit of cotangent function: We can use L'Hôpital's Rule to evaluate this limit because it is in the indeterminate form 0*(-∞). Differentiate the numerator and the denominator with respect to θ.Limit of natural logarithm is infinity: The derivative of the numerator ln(sin(θ)) with respect to θ is (1/sin(θ)) * cos(θ) = cot(θ), and the derivative of the denominator θ with respect to θ is 1.Limit does not exist: Applying L'Hôpital's Rule, we get lim_(θ→0^+)cot(θ)/1, which simplifies to lim_(θ→0^+)cot(θ).As θ approaches 0 from the positive side, cot(θ) approaches infinity. Therefore, the limit of ln(y) as θ approaches 0 from the positive side is infinity.Since the limit of ln(y) is infinity, the limit of y = e^(ln(y)) is e^(∞), which means the original limit lim_(θ→0^+)(sin(π - θ))^θ is e^(∞).However, e^(∞) is not a finite number; it represents that the function grows without bound. Therefore, the limit does not exist in the real number system.

$\lim _{\theta \rightarrow 0^{+}}(\sin (\pi-\theta))^{\theta}$

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lim⁡θ→0+(sin⁡(π−θ))θ \lim _{\theta \rightarrow 0^{+}}(\sin (\pi-\theta))^{\theta} limθ→0+​(sin(π−θ))θ

$\lim _{\theta \rightarrow 0^{+}}(\sin (\pi-\theta))^{\theta}$