Computing Limits Using Taylor's Formula

$\lim_{x \rightarrow 0} \frac{\ln (1+x)-x}{x^{2}}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {sh} 2 x-2 \operatorname {sh} x}{x^{3}}$;

$\lim_{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}}$;

$\lim_{x \rightarrow 0} \frac{\cos x-1+x^{2} / 2}{x^{4}}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {ch} 3 x+\cos 3 x-2}{x^{4}}$

$\lim_{x \rightarrow 0} \frac{\operatorname {tg} x-\sin x}{x^{3}}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+x}+\sqrt[3]{1+x}-2 \sqrt[4]{1-x}}{x}$

$\lim_{x \rightarrow 0} \frac{\operatorname {arctg} x-\arcsin x}{x^{2}}$.

$\lim_{x \rightarrow 0} \frac{\operatorname {tg} x-x}{\sin x-x}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {arctg} x-\arcsin x}{\operatorname {tg} x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{2 \arcsin x-\arcsin 2 x}{x^{3}}$;

$\lim_{x \rightarrow 0} \frac{\sqrt[5]{1+2 x}-1}{\sqrt[4]{1+x}-\sqrt{1-x}}$

$\lim_{x \rightarrow 0} \frac{1+x \cos x-\sqrt{1+2 x}}{\ln (1+x)-x}$;

$\lim_{x \rightarrow 0} \frac{e^{x}-\sqrt{1+2 x}}{\ln \cos x}$

$\lim_{x \rightarrow 0} \frac{3 \cos x+\arcsin x-3 \sqrt[3]{1+x}}{\ln \left(1-x^{2}\right)}$;

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{1-x^{2}}-x \operatorname {ctg} x}{x \sin x}$.

$\lim_{x \rightarrow 0} \frac{(1+x)^{1 / x}-e}{x}$;

$\lim_{x \rightarrow 0} \frac{(1+x)^{x}-1}{x^{2}}$;

$\lim_{x \rightarrow 0} \frac{(1+x)^{1 / x}-e(1-x / 2)}{x^{2}}$;

$\lim_{x \rightarrow 0} \frac{\cos ((\pi / 2) \cos x)}{\sin \left(\sin^{2} x\right)}$.

$\lim_{x \rightarrow 0} \frac{e^{x}-\sqrt[3]{1+3 x+9 x^{2} / 2}}{x^{3}}$;

$\lim_{x \rightarrow 0} \frac{\ln \left(1+x^{3}\right)-2 \sin x+2 x \cos x^{2}}{\operatorname {arctg} x^{3}}$;

$\lim_{x \rightarrow 0} \frac{x \sqrt{1+\sin x}-(1 / 2) \ln \left(1+x^{2}\right)-x}{\operatorname {tg}^{3} x}$;

$\lim_{x \rightarrow 0} \frac{e^{\sin x}-\sqrt{1+x^{2}}-x \cos x}{\ln^{3}(1-x)}$;

$\lim_{x \rightarrow 0} \frac{e^{\sin x \ln \cos x}-(1+4 x)^{1 / 4}+x-3 x^{2} / 2}{x \sin x^{2}}$.

$\lim_{x \rightarrow 0} \frac{e^{\operatorname {arctg} x}+\ln (1-x)-1}{2-\sqrt{4+x^{3}}}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+x} \sin x+\ln \cos x-x}{\sqrt[3]{1-x^{3}}-1}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+x} \operatorname {sh} x+\ln \cos x-x}{1-\sqrt[3]{1-x^{3}}}$;

$\lim_{x \rightarrow 0} \frac{\ln \left(e^{2 x}+\sin x\right)-3 \arcsin x+5 x^{2} / 2}{\sqrt[3]{8+x^{3}}-2}$;

$\lim_{x \rightarrow 0} \frac{\ln (1+\ln (1+x) /(1+x))-\operatorname {tg}\left(x-2 x^{2}\right)}{\sqrt{4+x^{3}}-2}$.

$\lim_{x \rightarrow 0} \frac{\cos x-\sqrt{1-x^{2}}}{\sin x-x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+2 x^{3}}-\cos x^{4}}{\operatorname {tg} x-x}$;

$\lim_{x \rightarrow 0} \frac{x \sqrt{1+\sin x}+\ln (1-x)}{\operatorname {tg} x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1-x} \ln (1+x)-x /(x+1)}{\operatorname {tg} x-\sin x}$.

$\lim_{x \rightarrow 0} \frac{e^{\sin x}+\ln (1-x)-1}{\arcsin x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+2 \operatorname {tg} x}-e^{x}+x^{2}}{\arcsin x-\sin x}$

$\lim_{x \rightarrow 0} \frac{\ln (1+x) \cos x-e^{\operatorname {tg} x}+\sqrt{1+2 x^{2}}}{x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1-\sin x}-\ln (1-x / 2)-1}{\operatorname {tg} x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{e^{\operatorname {tg} x}-x-\operatorname {ch} x}{\sin x-\operatorname {arctg} x}$;

$\lim_{x \rightarrow 0} \frac{e^{\sin x}+\ln (1-\sin x)-1}{\operatorname {tg} x-\operatorname {arctg} x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{1+3 x}-e^{\sin x}+3 x^{2} / 2}{\arcsin x-\operatorname {tg} x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{1+3 x} \ln (1-x)+\sin (\sin x)+3 x^{2} / 2}{\operatorname {tg} x-\arcsin x}$.

$\lim_{x \rightarrow 0} \frac{e^{\operatorname {tg} x}-\sqrt{1+2 x}-x\left(x+x^{2}\right)}{x-\operatorname {arctg} x}$;

$\lim_{x \rightarrow 0} \frac{\ln (1+x / 2)-\sqrt{1+\sin x}+1}{\operatorname {sh} x-\operatorname {arctg} x}$;

$\lim_{x \rightarrow 0} \frac{e^{\operatorname {tg}(x / 2)}-\sqrt{1+\sin x}-x^{2} / 4}{\arccos x-\operatorname {arcctg} x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{1+3 x+x^{2}}+\sin \ln (1-x)-e^{-7 x^{2} / 6}}{x-\operatorname {arctg} x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+\operatorname {sh} 2 x}-\cos x-x}{\operatorname {tg} x-\operatorname {arctg} \sin x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{2-e^{2 x}}-\cos 2 x+\ln (1+x)}{\sin x-\arcsin \operatorname {tg} x}$.

$\lim_{x \rightarrow 0} \frac{x e^{\operatorname {tg} x}-\sin^{2} x-x}{x+x^{3}-\operatorname {tg} x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{1+x^{3}}-x \operatorname {ctg} x-x^{2} / 3}{x \cos x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{\cos x-\sqrt{1-2 x}-x}{x^{2} \operatorname {tg} x-e^{-x^{3}}+1}$

$\lim_{x \rightarrow 0} \frac{\sin x-\ln \left(\sin x+\sqrt{1+x^{2}}\right)}{\operatorname {tg} x-x \cos^{2} x}$;

$\lim_{x \rightarrow 0} \frac{e^{e^{x}-1}-1 /(1-x)}{\ln ((1+x) /(1-x))-2 \sin x}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {tg}(\sin x)-\ln \left(x+\sqrt[3]{1+x^{2}}\right)-x^{2} / 6}{\operatorname {th}\left(x-x^{3}\right)-x}$.

$\lim_{x \rightarrow 0} \frac{e^{2 x}-\operatorname {ch} 2 x-2 x}{\operatorname {tg} 2 x-2 \sin x}$;

$\lim_{x \rightarrow 0} \frac{\ln \left(1+x-x^{2} / 6\right)-\operatorname {sh} x+2 x^{2} / 3}{\sin 2 x-2 x \cos x}$;

$\lim_{x \rightarrow 0} \frac{x^{2} e^{2 x}+\ln \left(1-x^{2}\right)}{x \cos x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{\arcsin x+3 \cos x-3 \sqrt[3]{1+x}}{1+\ln (1+x)-e^{x}}$;

$\lim_{x \rightarrow 0} \frac{x^{2} e^{x}-\ln \left(1+x^{2}\right)-\arcsin x^{3}}{x \sin x-x^{2}}$;

$\lim_{x \rightarrow 0} \frac{e^{1+\operatorname {tg} x}-e^{\sqrt{1+2 x}}}{\sin \left(x^{2} / 7\right)-(x / 3) \ln (1-x)}$;

$\lim_{x \rightarrow 0} \frac{e^{x}-\sqrt{1+2 x+2 x^{2}}}{x+\operatorname {tg} x-\sin 2 x}$

$\lim_{x \rightarrow 0} \frac{\operatorname {tg} \sin x-x \cos x}{e^{x}+\ln (1-x)-1}$.

$\lim_{x \rightarrow 0} \frac{e^{x}-x \sqrt{1+x}-1}{\sin x \operatorname {ch} x-\operatorname {sh} x}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {tg} x-\ln \left(x+\sqrt{1+x^{2}}\right)}{\sin x-x \cos x}$;

$\lim_{x \rightarrow 0} \frac{e^{x-x^{2}}-\ln (1+\sin x)-1}{x \cos x-\operatorname {sh} x}$;

$\lim_{x \rightarrow 0} \frac{2 \ln \cos x+x \operatorname {sh} x}{\sin \left(x^{2} / 2\right)-\operatorname {sh}\left(x^{2} / 2\right)}$

$\lim_{x \rightarrow 0} \frac{e^{x^{2}} \operatorname {tg} x-x \cos \sin x}{\ln (1+x)-x \sqrt{1-x}}$;

$\lim_{x \rightarrow 0} \frac{\arcsin x-x e^{x}}{x \sqrt{1-x^{2}}-\operatorname {tg} x}$

$\lim_{x \rightarrow 0} \frac{\sqrt{1-2 x}+e^{\operatorname {tg} x}-2}{\sin x / x-\cos x-x^{2} / 3}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{\cos x-x \ln (1+x)}+(3 / 4) \operatorname {tg} x^{2}-1}{x e^{x}-\arcsin x-x^{2}}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {sh} 2 x+\ln (1-\sin x)-\sin \ln (1+x)}{(1-2 x)^{-1 / 2}-e^{x}-x^{2}}$;

$\lim_{x \rightarrow 0} \frac{(1-2 x)^{-1 / 2}-(1+2 x)^{-1 / 2}-\operatorname {arctg} 2 x}{e^{-x}+\ln (1+\arcsin x)-1}$;

$\lim_{x \rightarrow 0} \frac{\ln \left(e^{\sin x}+\ln (1-x)+x^{3} / 3\right)}{\ln \operatorname {ch} x-x^{2} / 2}$;

$\lim_{x \rightarrow 0} \frac{\sin \operatorname {arctg} x-\operatorname {tg} x}{e^{\operatorname {sh} x}-(1+2 x)^{1 / 2}-x^{2}}$.

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{1+x^{2}}-e^{x^{3} / 4}}{\ln \left(1+3 x^{2}\right)-3 x^{2} \cos x}$;

$\lim_{x \rightarrow 0} \frac{\ln \left(\sqrt{1+x^{2}}-x\right)+\operatorname {tg} x}{x\left(\operatorname {ch} x-e^{x^{2}}\right)}$;

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{\ln (e+x)}-e^{x /(3 e)}+x^{2} /\left(3 e^{2}\right)}{x \operatorname {ch} x-\sin x}$;

$\lim_{x \rightarrow 0} \frac{e^{\sin x}-\sqrt{1+x^{2}}-\arcsin x}{\operatorname {sh}\left(x-x^{2}\right)-\ln \sqrt{1+2 x}}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {ch}\left(2 x /\left(2+x^{4}\right)\right)+\cos \left(2 x /\left(2-x^{4}\right)\right)-2 e^{x^{4} / 2}}{\operatorname {tg} \sqrt{1+x^{4}}-\operatorname {tg} \sqrt{1-x^{4}}}$;

$\lim_{x \rightarrow 0} \frac{e^{1+\cos x}-e^{2+x^{2}}+\left(3 e^{2} / 2\right) \sin x^{2}}{\ln \left(1+x^{2}\right)-(\operatorname {arctg} x)^{2}}$.

$\lim_{x \rightarrow 0} \frac{\ln (1+\sin 2 x)-2 x+2 x^{2}}{x / 2+\operatorname {tg}(x / 2)-\arcsin x}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+\sin x}-(1 / 2) \operatorname {tg} x+x^{2} / 8-1}{e^{x}-\sqrt{1+2 x}-x^{2}}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{1+2 x}-e^{\operatorname {tg} x}+6 x^{3}+x^{2}}{\ln (1+x)-\operatorname {arctg} x+x^{2} / 2}$;

$\lim_{x \rightarrow 0} \frac{\operatorname {ch} 2 x-(1+3 x)^{-1 / 3}-x}{x^{2} / 2+\ln (1+\operatorname {tg} x)-\arcsin x}$;

$\lim_{x \rightarrow 0} \frac{e^{x /(1-x)}-\operatorname {sh} x-\cos x}{\sqrt[6]{1+x}+\sqrt[6]{1-x}-2}$;

$\lim_{x \rightarrow 0} \frac{x+\operatorname {ch} x-e^{\arcsin x}}{\operatorname {tg} x+\sqrt[3]{1-3 x}-2 \cos x+1}$;

$\lim_{x \rightarrow 0} \frac{\sqrt{2 x+\cos 2 x}-e^{\operatorname {tg} x}+2 x^{2}}{2 \sin x-2 \ln (1+x)-x^{2}}$,

$\lim_{x \rightarrow 0} \frac{\operatorname {sh} \sin x^{3}+\sin \operatorname {sh} x^{3}}{\left(x^{2} / 2\right) \sqrt{1-x}+\ln (1+x)-x \cos x}$

$\lim_{x \rightarrow 0} \frac{\operatorname {tg}^{2} x-\operatorname {tg} x^{2}}{e^{\arcsin x}-e^{\sin x}-x^{3} / 2}$,

$\lim_{x \rightarrow 0} \frac{\ln (\sin x / x)+\operatorname {ch}(x / \sqrt{3})-1}{\operatorname {sh} x-\ln \left(x+\sqrt{1+x^{2}}\right)}$,

$\lim_{x \rightarrow 0} \frac{3 \operatorname {arctg} \sin x-\operatorname {tg} \operatorname {sh} 3 x}{\sqrt{1+x} \sin x^{3}-x^{2} \ln (1-16 x / 9)}$,

$\lim_{x \rightarrow 0} \frac{\sin \sqrt{1+x^{3}}-\sin 1}{\sqrt[5]{1-2 x \ln \cos x}-1}$

$\lim_{x \rightarrow 0} \frac{\operatorname {arctg}\left(3+x^{2}\right)-\operatorname {arctg}(2+\cos x)}{\ln (1+x)-e^{x}+1}$,

$\lim_{x \rightarrow 0} \frac{\arcsin \left(1 / 2+x^{2}\right)-\arcsin (\cos x-1 / 2)}{1+\ln \left(1+x^{2}\right)-\cos x}$

$\lim_{x \rightarrow 0} \frac{e^{x^{2}}-\sqrt{1+2 x^{2}}}{\operatorname {tg}^{4} x}$,

$\lim_{x \rightarrow 0} \frac{e^{x}+\ln (1-\sin x)-1}{\sqrt[3]{8-x^{4}}-2}$,

$\lim_{x \rightarrow 0} \frac{\ln \left(\cos x+x^{2} / 2\right)}{e^{-x^{2} / 2}-\cos x}$,

$\lim_{x \rightarrow 0} \frac{e^{x /(1+x)}-\cos \left(1-e^{-x}\right)-\operatorname {arctg} x}{x^{4}}$,

$\lim_{x \rightarrow 0} \frac{x \sqrt{e^{x}-\sqrt{1+2 x}-\sqrt[3]{x^{6}-x^{7}}}}{(1 / e)(1+x)^{1 / x}-\sqrt{1-x+7 x^{2} / 6}}$

$\lim_{x \rightarrow 0} \frac{x \sqrt{\ln (1+x)+\cos x+4 x^{3} / 3-\sqrt[3]{1+3 x}}}{\sqrt{1-x+x^{2} / 2}-(\cos x)^{1 / x}}$

$\lim_{x \rightarrow 0} \frac{\sin \left(x e^{x}\right)+\sin \left(x e^{-x}\right)-2 x-2 x^{3} / 3}{x^{5}}$,

$\lim_{x \rightarrow 0} \frac{\sin (x \cos x)+x \ln \left(1+2 x^{2} / 3\right)-x}{\sqrt{1+x^{2}}-1}$

$\lim_{x \rightarrow 0} \frac{\sqrt[3]{1-x^{2} / 2}-e^{-x^{2} / 6}}{x^{2} \ln (1+x)-\left(\operatorname {tg} x^{3}\right) \cos \operatorname {sh}(x / 2)}$

$\lim_{x \rightarrow 0} \frac{\ln (\cos x+x \sin x)-\left(x^{2} / 2\right) e^{x}}{(x / 2) \sqrt[3]{1-x}+\sqrt{1+x^{2} / 3}-\sin (x / 2)-1}$

$\lim_{x \rightarrow 0} \frac{e^{\cos x}-e \sqrt[3]{1-4 x^{2}}}{(1 / x) \arcsin 2 x-2 \operatorname {ch} x^{2}}$,

$\lim_{x \rightarrow 0} \frac{\ln (\sqrt{1+2 x}-\operatorname {tg} x)+(1 / 2) \operatorname {arctg} x^{2}}{x e^{x^{2}}-\sin x}$,

$\lim_{x \rightarrow 0} \frac{\sqrt{1-2 x}-e^{-x}+x^{2} \sqrt[3]{1+x}}{\sin^{2} x-\ln \operatorname {ch}^{2} x}$,

$\lim_{x \rightarrow 0} \frac{\ln (1+x)+(1 / 2) \operatorname {sh} x^{2}-x}{\sqrt{1+\operatorname {tg} x}-\sqrt{1+\sin x}}$

$\lim_{x \rightarrow 0} \frac{\left((1+3 x)^{1 / 3}-1\right) / \operatorname {tg} x-e^{-\operatorname {sh} x}-x^{2}(x+5) /(x+6)}{\ln \left(2 e^{x^{2}}-1\right) / \sin x-\operatorname {arctg} 2 x}$,

$\lim_{x \rightarrow 0} \frac{\left(\sqrt{\cos 2 x+\operatorname {sh} 2 x}-e^{x}\right) / x+2 x(2-x) /(2+x)}{(\ln \operatorname {ch} x) / \operatorname {sh} x-(1 / 2) \arcsin x}$.

$\lim_{x \rightarrow 0}(\sqrt{1+x}-x)^{1 / x}$;

$\lim_{x \rightarrow 0}(\cos x)^{\operatorname {ctg}^{2} x}$

$\lim_{x \rightarrow 0}(\operatorname {ch} x)^{1 / \sin^{2} x}$;

$\lim_{x \rightarrow 0}\left(\frac{\cos x}{\operatorname {ch} 3 x}\right)^{1 / x^{2}}$

$\lim_{x \rightarrow 0}\left(\frac{e^{x^{2}}}{\operatorname {ch} 3 x}\right)^{1 / x^{2}}$

$\lim_{x \rightarrow 0}\left(\ln (e+x)-\frac{x}{e}\right)^{1 / \sin^{3} x}$.

$\lim_{x \rightarrow 0}\left(\frac{\arcsin x}{x}\right)^{1 / x^{2}}$;

$\lim_{x \rightarrow 0}\left(\frac{\ln \left(\sqrt{1+x^{2}}+x\right)}{x}\right)^{1 / x^{2}}$

$\lim_{x \rightarrow 0}\left(\frac{\sin x}{\arcsin x}\right)^{1 / x^{2}}$;

$\lim_{x \rightarrow 0}\left(\frac{\operatorname {tg} x}{\operatorname {arctg} x}\right)^{1 / x^{2}}$.

$\lim_{x \rightarrow 0}\left(\frac{\sin x}{2(\sqrt{1+x}-1)}\right)^{\operatorname {ctg} x}$;

$\lim_{x \rightarrow 0}\left(\frac{\sqrt{1-2 x}-\sqrt[3]{1-3 x}}{\ln \operatorname {ch} x}\right)^{1 / x}$;

$\lim_{x \rightarrow 0}\left(\frac{1}{x} e^{x /(1+x)}-\frac{1}{\sin x}\right)^{1 / \operatorname {arctg} x}$;

$\lim_{x \rightarrow 0}\left(\frac{\operatorname {tg} 3 x+\cos 4 x-\cos 2 x}{\ln \sqrt{1+3 x}-\ln \sqrt{1-3 x}}\right)^{1 / \sin x}$.

$\lim_{x \rightarrow 0}\left(1+6 \frac{x-\sin x}{x^{2}}\right)^{2(\operatorname {ch} x-1) / x^{2}}$;

$\lim_{x \rightarrow 0}\left(\frac{e^{2}-(1+2 x)^{1 / x}}{2 x e^{2}}\right)^{1 / x}$;

$\lim_{x \rightarrow 0}\left(\frac{\operatorname {ch} x-\cos x}{2 \sqrt{1+2 x}-2 \sqrt[3]{1+3 x}}\right)^{1 / x}$.

$\lim_{x \rightarrow 0}(\sqrt{1+\operatorname {tg} 2 x}+\ln (1-x))^{1 / x^{2}}$;

$\lim_{x \rightarrow 0}(\operatorname {tg}(x / 3)+2-\sqrt[3]{1+x})^{\operatorname {ctg}^{2} x}$;

$\lim_{x \rightarrow 0}\left(e^{(1 / 3) \sin x}+\sqrt[3]{1-\operatorname {tg} x}-1\right)^{1 / \ln \left(1+x^{2}\right)}$;

$\lim_{x \rightarrow 0}\left(\frac{1}{e}(1+x)^{1 / x}+\frac{2 x}{4+5 x}\right)^{\operatorname {ctg}^{2} x}$.

$\lim_{x \rightarrow 0}\left(\frac{\operatorname {arctg} x}{e^{x}-1-x^{2} / 2}\right)^{1 / x^{2}}$;

$\lim_{x \rightarrow 0}\left(\frac{x \operatorname {sh} x}{\ln \left(1+x^{2}\right)}\right)^{\operatorname {ctg}^{2} x}$;

$\lim_{x \rightarrow 0}\left(\frac{2 \cos x+x}{2 \sqrt{1+x}}\right)^{1 / x^{2}}$;

$\lim_{x \rightarrow 0}\left(\frac{x \sin x}{2 \operatorname {ch} x-2}\right)^{1 / x^{2}}$.

$\lim_{x \rightarrow 0}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\operatorname {sh} x}\right)^{1 / \sin^{2} x}$;

$\lim_{x \rightarrow 0}\left(\frac{1+x^{2}-\sqrt{1+x^{2}}}{\operatorname {ch} x-1}\right)^{1 / x^{2}}$;

$\lim_{x \rightarrow 0}\left(\frac{\sqrt{\cos x}}{e^{x}-\ln (1+x)}\right)^{1 / x^{2}}$

$\lim_{x \rightarrow 0}\left(\frac{\sqrt{\cos x}}{\sqrt{1+\operatorname {tg} x^{2}}}\right)^{1 / x^{2}}$.

$\lim_{x \rightarrow 0}\left(\frac{(\arcsin x)^{2}-x^{2}}{\sin^{2}\left(x^{2} / \sqrt{3}\right)}\right)^{1 / \sin^{2} x}$;