Question Number 137157 by mathlove last updated on 30/Mar/21
Commented by mathlove last updated on 30/Mar/21
$${with}\:{out}\:{macloreen}\:{sirees}\:{and}\:{H}-{pital}\:{rools} \\ $$
Commented by mathlove last updated on 30/Mar/21
Commented by bobhans last updated on 30/Mar/21
$$\mathrm{L}'\mathrm{Hopital} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}{\mathrm{1}−\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}\right)}{\left(\mathrm{cos}\:\mathrm{x}−\mathrm{1}\right)\left(\mathrm{cos}\:\mathrm{x}+\mathrm{1}\right)} \\ $$$$=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by mathlove last updated on 30/Mar/21
$${weth}\:{out}\:\:\:{L}\:\:{Hopital}\:\:{rols} \\ $$
Commented by bobhans last updated on 30/Mar/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{3}} }\:.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}−\mathrm{tan}\:\mathrm{x}} \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{solve}\:\mathrm{it}\:\mathrm{without}\:\mathrm{L}'\mathrm{Hopital} \\ $$
Commented by mathlove last updated on 30/Mar/21
$${no}\:\:{I}\:\:{can}\:{not} \\ $$