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 $\ln 2 = \infty - \infty .$
$\ln 2 = \ln 1 + \ln 2 = \ln 1 - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty } \ln \left( \frac { b - 1 } { b } \right) - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty } \left[ \ln \frac { x - 1 } { x } \right] _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) - \ln x ] { } _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty } \int _ { 2 } ^ { b } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \frac { 1 } { x - 1 } d x - \int _ { 2 } ^ { \infty } \frac { 1 } { x } d x$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) ] \frac { b } { b } - \lim [ \ln x ]$
$= \infty - \infty$
$= \int _ { 2 } ^ { \infty } \frac { 1 } { x - 1 } d x - \int _ { 2 } ^ { \infty } \frac { 1 } { x } d x$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) ] _ { 2 } ^ { b } - \lim _ { b \rightarrow \infty } [ \ln x ] _ { 2 } ^ { b }$
$= \infty - \infty$

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Definitions:

Group Differences

The variations or disparities that exist between different groups, such as cultural, socioeconomic, or demographic distinctions.

Performance

The act of carrying out or accomplishing an action, task, or function.

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The risk of conforming to stereotypes about one's social group, which can hinder performance and behavior.

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The experience of feeling at risk of confirming a negative stereotype about one's social group, which can negatively affect performance and wellbeing.