lim _{x rightarrow 1} frac{a b^x-a^x b}{x^2-1 | vedclass

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(B) Let $L = \lim _{x \rightarrow 1} \frac{a b^x - a^x b}{x^2 - 1}$.Since the limit is of the form $\frac{0}{0}$ at $x = 1$,we apply $L$' Hospital's r

Vedclass · Accepted Answer

$\frac{ab}{2} \log \left(\frac{b}{a}\right)$

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(B) Let $L = \lim _{x \rightarrow 1} \frac{a b^x - a^x b}{x^2 - 1}$.Since the limit is of the form $\frac{0}{0}$ at $x = 1$,we apply $L$' Hospital's rule by differentiating the numerator and denominator with respect to $x$:$L = \lim _{x \rightarrow 1} \frac{\frac{d}{dx}(a b^x - a^x b)}{\frac{d}{dx}(x^2 - 1)}$$L = \lim _{x \rightarrow 1} \frac{a b^x \ln b - b a^x \ln a}{2x}$Substituting $x = 1$:$L = \frac{a b^1 \ln b - b a^1 \ln a}{2(1)}$$L = \frac{ab \ln b - ab \ln a}{2}$$L = \frac{ab}{2} (\ln b - \ln a)$$L = \frac{ab}{2} \ln \left(\frac{b}{a}\right)$

$\lim _{x \rightarrow 1} \frac{a b^x-a^x b}{x^2-1} = $

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