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Question Number 156024 by alcohol last updated on 07/Oct/21

Answered by mr W last updated on 07/Oct/21

lim_(x→0) ((√(1+f(3x)))/x)  =3lim_(x→0) ((√(1+f(3x)))/(3x))  =3lim_(t→0) ((√(1+f(t)))/t)  =3l

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}+{f}\left(\mathrm{3}{x}\right)}}{{x}} \\ $$$$=\mathrm{3}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}+{f}\left(\mathrm{3}{x}\right)}}{\mathrm{3}{x}} \\ $$$$=\mathrm{3}\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}+{f}\left({t}\right)}}{{t}} \\ $$$$=\mathrm{3}{l} \\ $$

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