Ucale

# 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

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

April 18, 2019
Which class you are presently in?
Choose an option. You can change your option at any time.
You will be solving questions and growing your critical thinking skills.