Evaluation of Logarithmic limits

In this lesson, we learn how to evaluate limits by using a logarithmic transformation(logarithmic limits). Several examples are included within the contents of the lesson.

$\displaystyle \left( i \right) \lim _{ x\longrightarrow 0 }{ \frac { { a }^{ x }-1 }{ x } } =\log _{ e }{ a } \\ =\lim _{ x\longrightarrow 0 }{ \frac { 1+x\log _{ e }{ a } +\frac { { x }^{ 2 } }{ 2! } { \left( \log _{ e }{ a } \right) }^{ 2 }+....-1 }{ x } } \\ =\lim _{ x\longrightarrow 0 }{ \log _{ e }{ a } +\frac { { x }^{ 2 } }{ 2! } { \left( \log _{ e }{ a } \right) }^{ 2 }+....= } \log _{ e }{ a }$

Putting x=e in the above result,we get $\displaystyle \lim _{ x\longrightarrow 0 }{ \frac { { e }^{ x }-1 }{ x } } =\log _{ e }{ a } =1$

$\displaystyle \left( i \right) \lim _{ x\longrightarrow 0 }{ \frac { \log { \left( 1+x \right) } }{ x } } \\ =\lim _{ x\longrightarrow 0 }{ \frac { x-\frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 3 } }{ 3 } -..... }{ x } } \\ =\lim _{ x\longrightarrow 0 }{ 1-\frac { x }{ 2 } +\frac { { x }^{ 3 } }{ 3 } ..... } =1$

Example

$\displaystyle \lim _{ x\longrightarrow 0 }{ \frac { { a }^{ x }-{ b }^{ x } }{ x } } \\ =\lim _{ x\longrightarrow 0 }{ \frac { \left( { a }^{ x }-1 \right) \left( { b }^{ x }-1 \right) }{ x } } \\ =\lim _{ x\longrightarrow 0 }{ \frac { \left( { a }^{ x }-1 \right) }{ x } } -\lim _{ x\longrightarrow 0 }{ \frac { \left( { b }^{ x }-1 \right) }{ x } } \\ =\log { a } -\log { b } =\log { \left( \frac { a }{ b } \right) }$