Provide an Appropriate Response $\ln 2 = \infty - \infty$ Where Does the Argument Go Wrong

question 155

Essay

Provide an appropriate response.
-Here is an argument that $\ln 2 = \infty - \infty$ . Where does the argument go wrong?
$\ln 2 = \ln 1 + \ln 2 = \ln 1 - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty } \ln \left( \frac { b - 1 } { b } \right) - \ln \frac { 1 } { 2 }$ $= \lim _ { b \rightarrow \infty } \left[ \ln \frac { x - 1 } { x } \right] _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) - \ln x ] _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty } \int _ { 2 } ^ { b } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
Here is an argument that $\ln 2 = \infty - \infty .$
$\ln 2 = \ln 1 + \ln 2 = \ln 1 - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty } \ln \left( \frac { b - 1 } { b } \right) - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty } \left[ \ln \frac { x - 1 } { x } \right] _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) - \ln x ] { } _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty } \int _ { 2 } ^ { b } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \frac { 1 } { x - 1 } d x - \int _ { 2 } ^ { \infty } \frac { 1 } { x } d x$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) ] \frac { b } { b } - \lim [ \ln x ]$
$= \infty - \infty$
$= \int _ { 2 } ^ { \infty } \frac { 1 } { x - 1 } d x - \int _ { 2 } ^ { \infty } \frac { 1 } { x } d x$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) ] _ { 2 } ^ { b } - \lim _ { b \rightarrow \infty } [ \ln x ] _ { 2 } ^ { b }$
$= \infty - \infty$

---

Definitions:

Added Value Negotiating

A negotiation technique where both parties work together to find solutions that enhance the value of the deal for everyone.

Openness

A trait or quality that involves being receptive to new ideas, experiences, and diverse perspectives, often considered crucial in creative and collaborative environments.

Commitment

A state or quality of being dedicated to a cause, activity, or goal, often demonstrated by persistent effort and determination.

Subsystem Adjustments

Modifications or changes made within a smaller part of an organization's system affecting its functioning or structure.