Only one of these calculations is correct. Which one? Why

Chapter 4, Problem 77E

(choose chapter or problem)

Only one of these calculations is correct. Which one? Why are the others wrong? Give reasons for your answers.

a. $\lim \limits _{x\rightarrow0^+}\ x\ \ln\ x=0\cdot(-\infty)=0$

b. $\lim \limits _{x\rightarrow0^+}\ x\ \ln\ x=0\cdot(-\infty)=-\infty$

c. $\lim \limits _{x\rightarrow0^+}\ x\ \ln\ x=\lim \limits _{x\rightarrow0^+}\ \frac{\ln\ x}{(1/x)}=\frac{-\infty}{\infty}=-1$

d. $\lim \limits _{x\rightarrow0^+}\ x\ \ln\ x=\lim \limits _{x\rightarrow0^+}\frac{\ln\ x}{(1/x)}$

$=\lim \limits _{x\rightarrow0^+}\ \frac{(1/x)}{\left(1/x^2\right)}=\lim \limits _{x\rightarrow0^+}\ (-x)=0$

Equation Transcription:

$\lim\limits_{{x} \rightarrow {{0}^{+}}}\ x\ ln\ x\ =\ 0\bullet{}(-\infty{})\ =\ 0$

$\lim\limits_{{x} \rightarrow {{0}^{+}}}\ x\ ln\ x\ =\ 0\bullet{}(-\infty{})\ =-\infty{}$

$\lim\limits_{{x} \rightarrow {{0}^{+}}}\ x\ ln\ x\ =\lim\limits_{{x} \rightarrow {{0}^{+}}}\frac{ln\ x}{(1/x)}=\ \frac{-\infty{}}{\infty{}}=-1$

$\lim\limits_{{x} \rightarrow {{0}^{+}}}x\ ln\ x\ =\lim\limits_{{x} \rightarrow {{0}^{+}}}\ \frac{ln\ x}{(1/x)}$

$=\lim\limits_{{x} \rightarrow {{0}^{+}}}\ \frac{(1/x)}{(1/x{\ }^{2})}=\lim\limits_{{x} \rightarrow {{0}^{+}}}\ (-x)=0$

Text Transcription:

lim_x rightarrow 0^+ x ln x = 0 cdot (- infty) = 0

lim_x rightarrow 0^+ x ln x = 0 cdot(- infty) = - infty

lim_x rightarrow 0^+ x ln x = lim_x rightarrow 0^+ frac ln x / (1/x)= frac - infty / infty = -1

lim_x rightarrow 0^+ x ln x = lim_x rightarrow 0^+ frac ln x / (1/x)

=lim_x rightarrow 0^+ frac (1/x) / (-1/x^2) =lim_x rightarrow 0^+ (-x)=0

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