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Question Number 27327 by Giannibo last updated on 05/Jan/18

lim_(n→∞) (x^(2n) /(1+∣x∣+x^(4n) ))

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{2n}} }{\mathrm{1}+\mid\mathrm{x}\mid+\mathrm{x}^{\mathrm{4n}} } \\ $$

Commented by abdo imad last updated on 05/Jan/18

= lim_(n−>∝)     ((/x/^(2n) )/(/x/^(4n) (   /x/^(−4n)  + /x/^(−3n) +1)))  =lim_(n−>∝  )   (1/(/x/^(2n) )).   (1/(/x/^(−4n) +/x/^(−3n)  +1))

$$=\:{lim}_{{n}−>\propto} \:\:\:\:\frac{/{x}/^{\mathrm{2}{n}} }{/{x}/^{\mathrm{4}{n}} \left(\:\:\:/{x}/^{−\mathrm{4}{n}} \:+\:/{x}/^{−\mathrm{3}{n}} +\mathrm{1}\right)} \\ $$$$={lim}_{{n}−>\propto\:\:} \:\:\frac{\mathrm{1}}{/{x}/^{\mathrm{2}{n}} }.\:\:\:\frac{\mathrm{1}}{/{x}/^{−\mathrm{4}{n}} +/{x}/^{−\mathrm{3}{n}} \:+\mathrm{1}} \\ $$

Commented by abdo imad last updated on 05/Jan/18

lim (....)=0

$${lim}\:\left(....\right)=\mathrm{0} \\ $$

Commented by prakash jain last updated on 05/Jan/18

Does it matter if ∣x∣<1 or ∣x∣>1  or ∣x∣=1?

$$\mathrm{Does}\:\mathrm{it}\:\mathrm{matter}\:\mathrm{if}\:\mid{x}\mid<\mathrm{1}\:\mathrm{or}\:\mid{x}\mid>\mathrm{1} \\ $$$$\mathrm{or}\:\mid{x}\mid=\mathrm{1}? \\ $$

Commented by abdo imad last updated on 05/Jan/18

yes yesif /x/<1   lim(...)=0  if /x/=1   lim(...)=(1/3)  i don t give attention to this case.

$${yes}\:{yesif}\:/{x}/<\mathrm{1}\:\:\:{lim}\left(...\right)=\mathrm{0} \\ $$$${if}\:/{x}/=\mathrm{1}\:\:\:{lim}\left(...\right)=\frac{\mathrm{1}}{\mathrm{3}}\:\:{i}\:{don}\:{t}\:{give}\:{attention}\:{to}\:{this}\:{case}. \\ $$

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