Evaluate the following limit using L'Hospital's rule. lim_(x rarr0^(+))(9x)^(sin(5x))

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Evaluate the following limit using L'Hospital's rule.  lim_(x rarr0^(+))(9x)^(sin(5x))

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Recognize Indeterminate Form: First, recognize that the limit is of the form 0^0, which is indeterminate. We can use L'Hospital's Rule by taking the natural log of the function and then exponentiating the result at the end.Take Natural Log: Let's set y = (9x)^(sin(5x)) and take the natural log of both sides to get ln(y) = sin(5x) * ln(9x).Find Limit of ln(y): Now we find the limit of ln(y) as x approaches 0 from the positive side. We have lim_(x rarr0^(+)) (sin(5x) * ln(9x)).Apply L'Hospital's Rule: We can apply L'Hospital's Rule to this limit since it's in the indeterminate form 0 * -∞. Differentiate the numerator and denominator with respect to x.Differentiate Numerator and Denominator: The derivative of sin(5x) with respect to x is 5cos(5x), and the derivative of ln(9x) with respect to x is 1/x.Simplify the Limit: Now we have lim_(x rarr0^(+)) (5cos(5x)/(1/x)). This simplifies to lim_(x rarr0^(+)) (5x * cos(5x)).Evaluate Limit: As x approaches 0, 5x approaches 0 and cos(5x) approaches 1. So the limit of 5x * cos(5x) as x approaches 0 is 0.Exponentiate to Find Limit of y: Since we took the natural log earlier, we need to exponentiate to get the limit of y. So, e^0 = 1.

Evaluate the following limit using L'Hospital's rule. $\newline$ $\lim _{x \rightarrow 0^{+}}(9 x)^{\sin (5 x)}$

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Evaluate the following limit using L'Hospital's rule.\newlinelim⁡x→0+(9x)sin⁡(5x) \lim _{x \rightarrow 0^{+}}(9 x)^{\sin (5 x)} x→0+lim​(9x)sin(5x)

Evaluate the following limit using L'Hospital's rule. $\newline$ $\lim _{x \rightarrow 0^{+}}(9 x)^{\sin (5 x)}$