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Question Number 85868 by M±th+et£s last updated on 25/Mar/20
if f(x)=⌊x^2 ⌋    and A=lim_(x→0) (f(x)−f(−x))  and B=f(x)+f(−x)  when x=0    find A and B
$${if}\:{f}\left({x}\right)=\lfloor{x}^{\mathrm{2}} \rfloor\:\: \\ $$$${and}\:{A}=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left({f}\left({x}\right)−{f}\left(−{x}\right)\right) \\ $$$${and}\:{B}={f}\left({x}\right)+{f}\left(−{x}\right)\:\:{when}\:{x}=\mathrm{0} \\ $$$$ \\ $$$${find}\:{A}\:{and}\:{B} \\ $$
Commented by mr W last updated on 25/Mar/20
A=0−0=0  B=0+0=0
$${A}=\mathrm{0}−\mathrm{0}=\mathrm{0} \\ $$$${B}=\mathrm{0}+\mathrm{0}=\mathrm{0} \\ $$
Commented by M±th+et£s last updated on 25/Mar/20
thank you sir sir but why lim_(x→0) (f(x)−f(−x))=0
$${thank}\:{you}\:{sir}\:{sir}\:{but}\:{why}\:\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left({f}\left({x}\right)−{f}\left(−{x}\right)\right)=\mathrm{0}\: \\ $$
Commented by M±th+et£s last updated on 25/Mar/20
sorry i  mean f(x)+f(−x)
$${sorry}\:{i}\:\:{mean}\:{f}\left({x}\right)+{f}\left(−{x}\right) \\ $$
Commented by M±th+et£s last updated on 25/Mar/20
⌊x⌋+⌊−x⌋= { ((0    x∈z)),((−1  x∉z)) :}  lim_(x→0^+ )  (−1)=−1  lim_(x→0^− )  (−1)=−1    so lim_(x→0) f(x)+f(−x)=−1
$$\lfloor{x}\rfloor+\lfloor−{x}\rfloor=\begin{cases}{\mathrm{0}\:\:\:\:{x}\in{z}}\\{−\mathrm{1}\:\:{x}\notin{z}}\end{cases} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {{lim}}\:\left(−\mathrm{1}\right)=−\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{−} } {{lim}}\:\left(−\mathrm{1}\right)=−\mathrm{1} \\ $$$$ \\ $$$${so}\:\underset{{x}\rightarrow\mathrm{0}} {{lim}f}\left({x}\right)+{f}\left(−{x}\right)=−\mathrm{1} \\ $$
Commented by mr W last updated on 25/Mar/20
but here f(x)=⌊x^2 ⌋ as you gave.
$${but}\:{here}\:{f}\left({x}\right)=\lfloor{x}^{\mathrm{2}} \rfloor\:{as}\:{you}\:{gave}. \\ $$
Commented by M±th+et£s last updated on 25/Mar/20
you are right sir iam so sorry it was a typo
$${you}\:{are}\:{right}\:{sir}\:{iam}\:{so}\:{sorry}\:{it}\:{was}\:{a}\:{typo} \\ $$

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