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Q. The value of $ \underset{x\to 0}{\mathop{\lim }}\,\frac{{{5}^{x}}-{{5}^{-x}}}{2x} $ is

JamiaJamia 2008

Solution:

$ \underset{x\to 0}{\mathop{\lim }}\,\frac{{{5}^{x}}-{{5}^{-x}}}{2x} $ Applying L- Hospitals rule $ =\underset{x\to 0}{\mathop{\lim }}\,\frac{{{5}^{x}}\log 5+{{5}^{-x}}\log 5}{2} $ $ =\frac{\log 5+\log 5}{2}=\log 5 $