Even and odd

Calculus Level 4

Given that $f(x)=\ln { x }$ and $g(x)=\dfrac { \ln { x } +\ln { \left( x+2 \right) } }{ \ln { \left( x+1 \right) } +\ln { \left( x+3 \right) } }$ , find $\displaystyle \lim _{ x\to { 0 }^{ + } }{ \frac { f\left( x \right) }{ g\left( x \right) } }$ .

\ln { 3 }

-\ln { 3 }

\ln { 4 }

\ln { \frac { 3 }{ 2 } }

Does not exist

1

1 solution

Chew-Seong Cheong
Mar 22, 2018

$\begin{aligned} L & = \lim_{x \to 0^+} \frac {f(x)}{g(x)} \\ & = \lim_{x \to 0^+} \frac {\ln x(\ln(x+1)+\ln(x+3))}{\ln x + \ln(x+2)} & \small \color{#3D99F6} \text{Divide up and down by} \ln x. \\ & = \lim_{x \to 0^+} \frac {\ln(x+1)+\ln(x+3)}{1 + \frac {\ln(x+2)}{\ln x}} \\ & = \frac {\ln 1 + \ln 3}{1+0} = \boxed{\ln 3} \end{aligned}$

Even and odd

1 solution

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