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Question Number 185438 by mathlove last updated on 21/Jan/23

$$\underset{x\to 0}{\lim }\frac{a^x\sin \left(ax\right)-b^x\sin \left(bx\right)}{c^x\sin \left(cx\right)-d^x\sin dx)}=?$$

Answered by hmr last updated on 21/Jan/23

$$L^{\prime }HopitalRule:$$$$\underset{x\to 0}{\lim }\frac{a^{x}ln\left(a\right)sin\left(ax\right)+a^{x+1}cos\left(ax\right)-b^{x}ln\left(b\right)sin\left(bx\right)-b^{x+1}cos\left(bx\right)}{c^{x}ln\left(c\right)sin\left(cx\right)+c^{x+1}cos\left(cx\right)-d^{x}ln\left(d\right)sin\left(dx\right)-d^{x+1}cos\left(dx\right)}$$$$=\frac{a-b}{c-d}$$$$$$

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