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lim-2-x-2-100-x-101x-




Question Number 184901 by mathlove last updated on 13/Jan/23
lim_((2/x)→2)  ((100+x)/(101x))
$$\underset{\frac{\mathrm{2}}{{x}}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{100}+{x}}{\mathrm{101}{x}} \\ $$
Commented by SEKRET last updated on 13/Jan/23
x→1
$$\boldsymbol{\mathrm{x}}\rightarrow\mathrm{1} \\ $$
Answered by cortano2 last updated on 13/Jan/23
lim_(t→2) ((100+(2/t))/(101((2/t))))=((101)/(101))=1
$${lim}_{{t}\rightarrow\mathrm{2}} \frac{\mathrm{100}+\frac{\mathrm{2}}{{t}}}{\mathrm{101}\left(\frac{\mathrm{2}}{{t}}\right)}=\frac{\mathrm{101}}{\mathrm{101}}=\mathrm{1} \\ $$
Answered by alephzero last updated on 13/Jan/23
lim_((2/x)→2) ((100+x)/(101x))  (2/x)→2  ⇒ x→1  ⇒ lim_((2/x)→2) ((100+x)/(101x)) = lim_(x→1) ((100+x)/(101x)) =  = ((100+1)/(101 ∙ 1)) = ((101)/(101)) = 1
$$\underset{\frac{\mathrm{2}}{{x}}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\mathrm{100}+{x}}{\mathrm{101}{x}} \\ $$$$\frac{\mathrm{2}}{{x}}\rightarrow\mathrm{2} \\ $$$$\Rightarrow\:{x}\rightarrow\mathrm{1} \\ $$$$\Rightarrow\:\underset{\frac{\mathrm{2}}{{x}}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\mathrm{100}+{x}}{\mathrm{101}{x}}\:=\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\mathrm{100}+{x}}{\mathrm{101}{x}}\:= \\ $$$$=\:\frac{\mathrm{100}+\mathrm{1}}{\mathrm{101}\:\centerdot\:\mathrm{1}}\:=\:\frac{\mathrm{101}}{\mathrm{101}}\:=\:\mathrm{1} \\ $$

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