Dirty Limits

Calculus Level 4

$\lim _{ x\rightarrow \infty }{ \frac { \ln { \left( x+\ln { \left( x+\ln { \left( { x }^{ \frac { 1 }{ x } } \right) } \right) } \right) } }{ \log _{ 10 }{ \left( { x }^{ 2 } \right) } } }$

This expression can be expressed as $\frac { A }{ B } \left( \ln { \left( C \right) } +\ln { \left( D \right) } \right)$ for co-prime positive integers $A$ and $B$ , and prime numbers $C$ and $D$ .

Find $A+B+C+D$

The answer is 10.

1 solution

Raj Rajput
Apr 29, 2015

In denominator change base and simply using property of log i.e,log(x square ) as 2log(x) further i was not getting any idea so i applied evergreen method L'Hospital simplified further i saw some coming in 1/x type form and applying limits to them =o then i remain with

numerator=1/( x + ln(x+1/x(lnx) )

denominator = 2/x

multiplying both of above by x gives us half ans 1/2 and as we have changed limits we have separately log10 to base e in multiplication

and answer comes to =1/2 ( log10) now 10 can be written as 2x5 or 1x10

getting values of A=1 B=2 C=2 D=5 summing up results A+B+C+D= 10 that is final answer ............ (answer can't be 14 as D=1 is not possible given D is a prime number :) :) )

Answer can't be 14 because D can't be 1. It is given that C and D are prime numbers, and 1 is not a prime.

Archit Boobna - 6 years, 1 month ago

thanks i just forget that point actually i forget to read it

RAJ RAJPUT - 6 years, 1 month ago

Is your profile picture showing the face of the protagonist of the show "Mahakumbh"?

Archit Boobna - 6 years, 1 month ago

Dirty Limits

The answer is 10.

1 solution

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