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Question
\[\lim_{x \to 0} \frac{e^{bx} - e^{ax}}{x} \text{ where } 0 < a < b\]
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Solution
\[\lim_{x \to 0} \left[ \frac{e^{bx} - e^{ax}}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \left( \frac{e^{bx} - 1}{x} \right) - \left( \frac{e^{ax} - 1}{x} \right) \right]\]
\[ = \lim_{x \to 0} \left[ b\left( \frac{e^{bx} - 1}{bx} \right) - a \times \left( \frac{e^{ax} - 1}{ax} \right) \right]\]
\[ = b \times 1 - a \times 1\]
\[ = b - a\]
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