Menu Close

lim-x-0-x-sin-x-x-n-n-N-




Question Number 195021 by tri26112004 last updated on 22/Jul/23
lim_(x→0)  ((x−sin x)/x^n )=¿ (n∈N^∗ )
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−{sin}\:{x}}{{x}^{{n}} }=¿\:\left({n}\in{N}^{\ast} \right) \\ $$
Answered by MM42 last updated on 22/Jul/23
if  x→0 ⇒ x−sinx ∼ (1/6)x^3   ⇒if   n<3 ⇒ lim=0  if  n=3  ⇒lim=(1/6)  if  n>3 ⇒ lim=∞
$${if}\:\:{x}\rightarrow\mathrm{0}\:\Rightarrow\:{x}−{sinx}\:\sim\:\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} \\ $$$$\Rightarrow{if}\:\:\:{n}<\mathrm{3}\:\Rightarrow\:{lim}=\mathrm{0} \\ $$$${if}\:\:{n}=\mathrm{3}\:\:\Rightarrow{lim}=\frac{\mathrm{1}}{\mathrm{6}} \\ $$$${if}\:\:{n}>\mathrm{3}\:\Rightarrow\:{lim}=\infty \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *