3. Reguła de l'Hospitala

Zadania

1

Stosując regułę de l'Hospitala oblicz granice funkcji:

  1. \lim\limits_{x\to+\infty}\frac {e^{3x}}{x+3},
  2. \lim\limits_{x\to+\infty}\frac{x^2+1}{\ln x},
  3. \lim\limits_{x\to0}\frac{\sin x}{e^x-1},
  4. \lim\limits_{x\to0}\frac{\mathrm{arctg}x}{\arcsin x},
  5. \lim\limits_{x\to4^-}\frac{\mathrm{arctg}(x-4)}{\sqrt x-2},
  6. \lim\limits_{x\to1^+}\frac{\ln(2-x)}{(x-1)^2},
  7. \lim\limits_{x\to0}\frac{x-\sin x}{x-\mathrm{tg} x},
  8. \lim\limits_{x\to-\infty}\frac{\sin\frac1x}{1-e^{\frac1x}},
  9. \lim\limits_{x\to0}\frac{x^2}{1-\sqrt{\cos x}}.
  1. +\infty,
  2. +\infty,
  3. 1,
  4. 1,
  5. 4,
  6. -\infty,
  7. -\frac{1}{2},
  8. -1,
  9. 4.
2

Oblicz granice funkcji:

  1. \lim\limits_{x\to+\infty}\left(e^x-\ln x\right),
  2. \lim\limits_{x\to0^+}\left(\frac1x-\frac{1}{\mathrm{tg} x}\right),
  3. \lim\limits_{x\to-\infty}\left(x^2+1\right)\cdot e^x,
  4. \lim\limits_{x\to0^+}\frac1{x\ln x},
  5. \lim\limits_{x\to0^+}\left(1-e^x\right)\cdot\mathrm{ctg} x,
  6. \lim\limits_{x\to0^-}x\cdot e^{-\frac1x},
  7. \lim\limits_{x\to0^+}\left(\sin x\right)^x,
  8. \lim\limits_{x\to0^+}x^{\sin x}.
  1. \left(e^x-\ln x\right)= e^x\cdot \left(1-\frac{\ln x}{e^ x}\right),
  2. \left(\frac1x-\frac{1}{\mathrm{tg} x}\right)=\frac{\mathrm{tg} x-x}{x\mathrm{tg} x},
  3. \left(x^2+1\right)\cdot e^x= \frac{x^2-1}{e^{-x}},
  4. x\cdot\ln x=\frac{\ln x}{\frac{1}x},
  5. \left(1-e^x\right)\cdot\mathrm{ctg} x=\frac{1-e^x}{\mathrm{tg} x},
  6. x\cdot e^{-\frac1x}=\frac{e^{-\frac1x}}{\frac1{x}},
  7. \left(\sin x\right)^x=e^{x\ln (\sin x)},\; x\cdot\ln (\sin x)=\frac{\ln (\sin x)}{\frac{1}x},
  8. x^{\sin x}=e^{\sin x\cdot\ln x},\; \sin x\cdot\ln x=\frac{\ln x}{\frac{1}{\sin x}}.
  1. +\infty,
  2. +\infty,
  3. 1,
  4. 1,
  5. 4,
  6. -\infty,
  7. -\frac{1}{2},
  8. -1,
  9. 4.