Problems

its graph has an asymptote as $x \rightarrow +\infty$, but $\lim_{x \rightarrow +\infty } f^{\prime }(x)$ does not exist;

its graph has no asymptote as $x \rightarrow +\infty$, but $\lim_{x \rightarrow +\infty } f^{\prime }(x)$ exists.

$y = x^{3}-3 x^{2}+4$;

$y = -x^{3}+4 x-3$;

$y = (x-1)^{2}(x+2)$;

$y = \frac{x^{2}}{4}-3 x+4$

$y = x(x-1)^{3}$

$y = (x+2)^{2}(x-1)^{2}$;

$y = (x-1)^{3}(x+1)^{2}$;

$y = 32 x^{2}\left(x^{2}-1\right)^{3}$.

$y = \frac{x^{2}+x-1}{x^{2}-2 x+1}$

$y = \frac{4+x-2 x^{2}}{(x-2)^{2}}$

$y = \frac{20 x^{2}}{(x-1)^{3}}$

$y = \frac{(x-1)^{2}}{(x+1)^{3}}$

$y = \frac{x^{3}}{x-1}$

$y = \frac{x^{3}-2 x^{2}-x+2}{x}$;

$y = \frac{1+x^{2}}{1+(x-2)^{2}}$;

$y = \frac{5 x^{2}+42 x+77}{x^{2}+7 x+14}$.

$y = \frac{x^{3}}{x^{2}-1}$;

$y = \frac{(x-1)^{3}}{(x-2)^{2}}$

$y = \frac{(x-5)^{3}}{(x-7)^{2}}$

$y = \frac{x^{3}+2 x^{2}}{(x-1)^{2}}$

$y = x+\frac{7}{x}-\frac{3}{x^{2}}$;

$y = (x+1)\left(\frac{x-1}{x-2}\right)^{2}$.

$y = \frac{x^{4}}{x^{3}+2}$;

$y = \frac{x^{4}}{(x+1)^{3}}$;

$y = 3 x+\frac{6}{x}-\frac{1}{x^{3}}$;

$y = \left(\frac{x+1}{x-1}\right)^{4}$;

$y = \frac{x^{5}}{\left(x^{2}-1\right)^{2}}$;

$y = \frac{(x-1)^{5}}{(x-2)^{4}}$

$y = \frac{x^{5}-8}{x^{4}}$;

$y = \frac{x^{5}}{x^{4}-1}$.

$y = x+\sqrt{x^{2}-1}$;

$y = x-\sqrt{x^{2}-2 x}$;

$y = \sqrt[3]{(x+1)^{2}}+\sqrt[3]{(x-1)^{2}}$;

$y = \sqrt[3]{(x+1)^{2}}-\sqrt[3]{(x-2)^{2}}$;

$y = \sqrt{x^{2}+1}-2 \sqrt{x+1}$;

$y = \sqrt{(2 x+1)^{3}} / 3+4 \sqrt{x}$.

$y = \sqrt{2 x^{3}+9 x^{2}}$;

$y = \sqrt{x^{2}-x^{3}}$;

$y = \sqrt{x^{3}-3 x}$;

$y = x^{2} \sqrt{x+1}$;

$y = x(x+1)^{3 / 2}$;

$y = \sqrt[4]{x^{4}-4 x^{3}}$.

$y = \frac{x+2}{\sqrt{x^{2}+2}}$;

$y = \frac{x+8}{\sqrt{x^{2}+4 x+16}}$

$y = \frac{8 x}{\sqrt{x^{2}-4}}$

$y = \frac{\sqrt{4 x^{2}-1}}{x}$

$y = \frac{\sqrt{x^{2}-4 x}}{2-x}$

$y = \frac{x^{2} \sqrt{x^{2}-1}}{2 x^{2}-1}$

$y = \frac{3 x-2}{\sqrt{x^{2}-1}}$;

$y = \sqrt{\frac{(x+6)^{2}}{x^{2}-4}}$

$y = 4 \sqrt{\frac{(x-1)^{2}}{x^{3}}}$

$y = \sqrt{\frac{3 x^{2}-4}{x^{3}}}$

$y = \sqrt{\frac{x^{2}}{3}-\frac{2}{3 x}}$

$y = \frac{1}{3} \sqrt{\frac{x^{3}}{x-2}}$

$y = \frac{4 x}{\sqrt{x^{2}+1}}-\frac{x}{2}$.

$y = \sqrt[3]{1-x^{3}}$;

$y = \sqrt[3]{x^{2}(3-x)}$;

$y = \sqrt[3]{x(x-1)^{2}}$;

$y = \sqrt[3]{x^{3}-4 x}$;

$y = x \sqrt[3]{(x-5)^{2}}$;

$y = (x+1)^{3} \sqrt[3]{(x-1)^{2}}$;

$y = (1+x) x^{2 / 3}$;

$y = x^{3}(x-1)^{2 / 3}$;

$y = \left(x^{2}-4\right)^{2 / 3}$;

$y = \left(x^{2}+8 x+12\right)^{2 / 3}$;

$y = \sqrt[3]{x(3-x)^{2}}-x$;

$y = \sqrt[3]{x^{2}}-\sqrt[3]{x^{2}-4}$

$y = \frac{x}{\sqrt[3]{x^{2}-1}}$;

$y = \frac{x}{\sqrt[3]{x+1}}$

$y = \frac{x}{\sqrt[3]{(x-2)^{2}}}$

$y = \sqrt[3]{\frac{x^{2}}{1+x}}$

$y = \sqrt[3]{\frac{(3 x-2)^{2}}{x-1}}$

$y = \sqrt[3]{\left(\frac{x+1}{x+2}\right)^{2}}$

$y = \frac{\sqrt[3]{x^{2}}}{x+2}$;

$y = \frac{\sqrt[3]{(x+1)^{2}}}{x^{2}}$.

$y = \left|x\right| \sqrt{1-x^{2}}$;

$y = x \sqrt{\left|x^{2}-1\right|}$

$y = 4 \frac{\sqrt{\left|x-1\right|}}{x-2}$;

$y = \sqrt{\left|3 x^{2}-x^{3}\right|}$;

$y = (x+1) \sqrt{\left|x^{2}-1\right|}$

$y = \frac{\sqrt{1+\left|x-2\right|}}{1+\left|x\right|}$

$y = \left(x^{2}-1\right) \sqrt{x+1}$;

$y = \frac{\sqrt{\left|x\right|-1}}{x-2}$

$y = \left|x\right| \sqrt[3]{1+3 x}$

$y = \sqrt[3]{x^{2}\left|2-x\right|}$.

$y = e^{x}-x$;

$y = x e^{-2 x}$;

$y = x^{2} e^{-x}$;

$y = x^{3} e^{-x}$;

$y = \left(x^{2}-2\right) e^{-2 x}$;

$y = (1-x) e^{3 x+1}$;

$y = e^{1-x^{2}}$;

$y = e^{4 x-x^{2}}$;

$y = x e^{-x^{2} / 2}$;

$y = \left(x^{2}+2\right) e^{-x^{2}}$;

$y = \frac{e^{-x}}{1-x}$.

$y = e^{(1-x) /(1+x)}$;

$y = x^{2} e^{1 / x}$

$y = (x-2) e^{-1 / x}$

$y = \frac{x^{2}+2 x-3}{x} e^{1 / x}$;

$y = x e^{1 / x^{2}}$.

$y = \ln x-x+1$;

$y = \frac{\ln x}{x}$;

$y = \frac{\ln x}{\sqrt{x}}$

$y = x^{2} \ln x$

$y = x \ln^{2} x$

$y = \frac{\ln^{2} x}{x}$;

$y = \frac{x}{\ln x}$

$y = \ln \left|\frac{x-1}{x+1}\right|+\frac{6}{x+1}$

$y = x^{2}-2 \ln x$.

$y = \cos x+\frac{1}{2} \sin 2 x$;

$y = \sin x+\frac{1}{2} \sin 2 x$;

$y = \sin x-\sin^{2} x$;

$y = \cos x-\frac{1}{2} \cos 2 x$;

$y = \cos 3 x+3 \cos x$.

$y = \sin x \sin 3 x$;

$y = \cos x \cos 2 x$;

$y = \sin x+\frac{1}{2} \sin 2 x+\frac{1}{3} \sin 3 x$.

$y = \frac{\cos 2 x}{\cos x}$;

$y = \frac{\sin (x-\pi / 4)}{\sin x}$;

$y = 2 x-\operatorname {tg} x$.

$y = \frac{x}{2}-\operatorname {arctg} x$;

$y = \frac{1}{\operatorname {arcctg} x}$

$y = x \operatorname {arctg} x$

$y = \frac{x}{2}+2 \operatorname {arcctg} x$

$y = \frac{3}{2} x-\arccos \frac{1}{x}$

$y = \arcsin \frac{2 x}{1+x^{2}}$

$y = \arccos \frac{1-x^{2}}{1+x^{2}}$

$y = \frac{x}{2}-\arccos \frac{2 x}{1+x^{2}}$.

$y = e^{\cos x}$;

$y = e^{-\operatorname {arctg} x}$;

$y = \sin x-\ln \sin x$;

$y = x^{x}$

$y = (1+x)^{1 / x}$

$y = (1+1 / x)^{x}$.

$y = x^{1 / x}$;

$y = x(1+1 / x)^{x}, x > 0$;

$y = \cos^{3} x+\sin^{3} x$;

$y = \sin 5 x-5 \sin x$

$y = \frac{\sin^{2} x}{2-\sin x}$;

$y = \cos x+\frac{\cos 2 x}{2}+\frac{\cos 3 x}{3}$;

$y = 2 \ln x-5 \operatorname {arctg} x$;

$y = \frac{1}{1+x^{2}} e^{1 /\left(1-x^{2}\right)}$;

$y = \frac{x^{2}}{x^{2}-4} e^{1 / x}$

$x = t^{3}+3 t+1, y = t^{3}-3 t+1$;

$x = t^{3}-3 \pi , y = t^{3}-6 \operatorname {arctg} t$;

$x = \frac{t^{3}}{1+t^{2}}, y = \frac{t^{3}-2 t^{2}}{1+t^{2}}$

$x = \ln \sin (t / 2), y = \ln \sin t$

$x = t-\sin t, y = 1-\cos t$ (cycloid);

$x = \cos t+\ln \operatorname {tg}(t / 2), y = \sin t$ (tractrix).

$x = t^{3}+2 t^{2}+t, y = -2+3 t-t^{3}$

$x = (t-1)^{2}(t-2), y = (t-1)^{2}(t-3)$;

$x = \frac{1}{t(t+1)}, y = \frac{(t+1)^{2}}{t}$;

$x = \frac{t^{2}}{t-1}, y = \frac{t^{2}-1}{t}$;

$x = \frac{(t+1)^{2}}{t}, y = \frac{t+1}{t+2}$

$x = \frac{t^{2}}{t^{2}-1}, y = \frac{t^{2}+1}{t+2}$

$x = \frac{t^{2}+1}{t}, y = \frac{t^{3}+1}{t^{2}}$.

$x = \frac{t^{2}+6 t+5}{3}, y = \frac{t^{3}-54}{2 t}$;

$x = \frac{t^{2}}{1-2 t}, y = \frac{t^{3}}{1-2 t}$;

$x = \frac{t^{2}}{1+t^{3}}, y = \frac{t^{3}}{1+t^{3}}$;

$x = t^{3}-3 t, y = \left(\frac{t-1}{t}\right)^{2}$;

$x = \frac{1}{t-t^{2}}, y = \frac{1}{t-t^{3}}$;

$x = \frac{1}{t^{3}-t^{2}}, y = \frac{1}{t^{2}-t}$;

$x = \frac{1}{t-t^{5}}, y = \frac{t^{4}}{1-t^{4}}$

$x = \sqrt{2} \frac{t+t^{3}}{1+t^{4}}, y = \sqrt{2} \frac{t-t^{3}}{1+t^{4}}$.

$x = e^{t}-t, y = e^{2 t}-2 t$;

$x = t e^{t}, y = t e^{-t}$;

$x = \left(t^{3}-2 t^{2}+3 t-4\right) e^{t}, y = \left(t^{3}-2 t^{2}+4 t-4\right) e^{t}$;

$x = e^{t} / t, y = (t-1)^{2} e^{t}$

$x = e^{t} /(t+1), y = e^{-t} /(t+1) x$;

$x = 2 t+\ln \left|t-1\right|, y = t+\ln \left|t-1\right|$;

$x = t \ln t, y = \frac{\ln t}{t}$;

$x = 2 t^{2}, y = \frac{t^{2}}{2}-3 \ln \left|\frac{t-1}{t+1}\right|$;

$x = \frac{\sin t+\cos t}{\sin t}, y = \frac{\cos 2 t}{\sin t}$;

$x = \operatorname {ctg} 2 t, y = \frac{2 \cos 2 t-1}{2 \cos t}$

$x = 2 \cos t-\cos 2 t, y = 2 \sin t-\sin 2 t$;

$x = 2 \cos 2 t, y = 2 \cos 3 t$;

$x = \sin 2 t, y = \sin 3 t$;

$x = \cos t+t \sin t, y = \sin t-t \cos t, t \geqslant 0$;

$x = e^{t} \cos t, y = e^{t} \sin t$.

$x^{3}-y^{3} = 1$;

$x^{4}+y^{4} = 1$;

$y^{2}(1-x) = x^{2}(1+x)$;

$3 y^{2} x = x^{3}-2$;

$y^{2} = 2 x^{3}-x^{4}$;

$y^{2} = 9\left(x^{4}-x^{6}\right)$;

$y^{2} x^{2} = 4(x-1)$;

$y^{2}(2-x) = x^{3}$;

$y^{2} x^{4} = \left(x^{2}-1\right)^{3}$;

$y^{2}\left(x^{2}-1\right) = x^{4}-4 x^{2}$;

$(x-1)\left(y^{2}-x^{2} / 3\right) = 4 x^{2} / 3$.

$(x-y+1)(x+y-1) = 1$;

$(x-2 y)^{2}+(4 x+2 y)^{2} = 4$;

$x^{2} y^{2}+y = 1$;

$x y^{2}+x^{2} y = 1$;

$x y(x-y)+x+y = 0$;

$x^{3}+y^{3} = 6 x^{2}$.

$x^{4}-y^{4} = 4 x^{2} y$;

$(x+y)^{3} = x y$;

$(x+y)^{4} = x^{2}+y^{2}$;

$x^{4}-2 x^{2} y^{2}+y^{3} = 0$;

$x^{3}-y^{3}+2 x-y = 0$;

$\left(x^{2}-y^{2}\right)(x-y) = 1$;

$x^{2 / 3}+y^{2 / 3} = 1$;

$x^{4 / 3}-y^{4 / 3} = 1$.

$\left(x^{2}+y^{2}\right) x = y$;

$\left(x^{2}+y^{2}\right)^{2} = x y$;

$x^{4}+y^{4} = x^{2}+y^{2}$;

$x^{4}+y^{4} = 2 x y$;

$x^{4}-y^{4} = x y$;

$x^{4}-y^{4} = x^{2}-2 y^{2}$;

$\left(x^{2}+y^{2}-2 x\right)^{2} = x^{2}+y^{2}$;

$\left(x^{2}+y^{2}-x\right)^{2} = 4\left(x^{2}+y^{2}\right)$.

$x^{4}+y^{4}-6 y^{3}+8 x^{2} y = 0$;

$\left(x^{2}+y^{2}\right)^{3} = 27 x^{2} y^{2}$

$x^{4}+2 y^{3} = 4 x^{2} y$;

$\left(x^{2}-y^{2}\right)(x-y) = 4 x^{2}$;

$x^{2} y^{2}+y^{4} = 4 x^{2}$;

$x^{3}+y^{3} = x^{2}+y^{2}$;

$x^{2} y^{2} = x^{3}-y^{3}$;

$x^{y} = y^{x}$.

$r = \left|\sin 2 \varphi \right|$;

$r = \cos 3 \varphi$;

$r = \operatorname {tg} 2 \varphi$;

$r = 1 / \sqrt{\sin 3 \varphi }$;

$r = 2+\cos \varphi$;

$r = 1+\cos \varphi$;

$r = 1+2 \cos \varphi$;

$r = 1-2 \cos \varphi$;

$r = (2 / \cos \varphi )-1$;

$r = 1+\operatorname {tg} \varphi$.