A) \[1\]
B) \[2\]
C) \[4\]
D) \[5\]
Correct Answer: A
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{e}^{5x}}-{{e}^{4x}}}{x}\] \[=\underset{x\to 0}{\mathop{\lim }}\,\,\,\frac{\left[ \begin{align} & \left( 1+\frac{5x}{1!}+\frac{{{(5x)}^{2}}}{2!}+.... \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ & -\left( 1+\frac{4x}{11}+\frac{{{(4x)}^{2}}}{2!}+..... \right) \\ \end{align} \right]}{x}\] \[=\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\,\frac{x\left[ \left( \frac{5}{1!}+\frac{25x}{2!}+..... \right)-\left( \frac{4}{1!}+\frac{16x}{2!}+.... \right) \right]}{x}\] \[=1\]
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