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Q.
$\underset{ h \rightarrow 0}{\text{Lim}} \frac{( e + h )^{\ln ( e + h )}- e }{ h }$ is
Continuity and Differentiability
Solution:
Limit is $f ^{\prime}( e )$ where $f ( x )= x ^{\ln x }= e ^{\ln ^2 x }$
$\Rightarrow f ^{\prime}( x ) = e ^{\ln ^2 x } \cdot \frac{2 \ln x }{ x } $
$f ^{\prime}( e ) = e \cdot \frac{2}{ e }=2$