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Derivabilitate

1. Sa se calculeze derivata functiei:

1.1. f(x) = x^{3} + x^{2} − x + 1.  1.2. f(x) = \frac{1}{4} x^{4} − \frac{1}{3} x^{3} + 2x^{2} − 1.
1.3. f(x) = 2x^{5} − x^{−2} + 3x.  1.4. f(x) = \frac{1}{x} − \frac{4}{x^{2}} − \frac{1}{x^{3}} + x.
1.5. f(x) = x^{\frac{1}{2}} + x^{\frac{2}{3}} − x^{− \frac{1}{3}} .  1.6. f(x) = \sqrt[3]{x} − \frac{1}{\sqrt[3]{x}} + 1.
1.7. f(x) = e^{x} sin x.  1.8. f(x) = tg x ln x.
1.9. f(x) = 2^{x} ctg x.  1.10. f(x) = x arcsin x.
1.11. f(x) = (x^{2} + 1) arctg x.  1.12. f(x) = x^{2} ln x.
1.13. f(x) = cos x ln x.  1.14. f(x) = x arcctg x.
1.15. f(x) = \frac{x}{x^{2} − 1}.  1.16. f(x) = \frac{x^{2} − 1}{x^{2} + 1}.
1.17. f(x) = \frac{sin x}{ln x}.  1.18. f(x) = \frac{arctg x}{e^{x}}.
1.19. f(x) = \frac{sin x − cos x}{sin x + cos x}.  1.20. f(x) = \frac{1 − sin x}{1 + sin x}.
1.21. f(x) = ln 3 − cos 2.  1.22. f(x) = arcsin x + arccos x.

2. Sa se calculeze derivata functiei:

2.1. f(x) = (x^{2} + 1)^{10}.  2.2. f(x) = \frac{1}{(x^{2} + 2x + 3)^{3}} .
2.3. f(x) = \sqrt{x^{2} − x + 7}.  2.4. f(x) = \frac{1}{\sqrt[3]{x^{3} + x^{2} + 1}}.
2.5. f(x) = sin^{2} x.  2.6. f(x) = ln^{2} x.
2.7. f(x) = sin 3x.  2.8. f(x) = sin (ln x).
2.9. f(x) = cos 2x.  2.10. f(x) = cos (e^{x}).
2.11. f(x) = tg 3x.  2.12. f(x) = tg 2^{x}.
2.13. f(x) = ctg \frac{x}{2}.  2.14. f(x) = ctg(x^{2} + x + 1).
2.15. f(x) = e^{sin x}.  2.16. f(x) = e^{−x}.
2.17. f(x) = 2^{tg x}.  2.18. f(x) = 3^{\sqrt{x}}.
2.19. f(x) = ln (sin x).  2.20. f(x) = ln (arctg x).

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2.21. f(x) = arctg \sqrt{x}.  2.22. f(x) = arctg e^{x}.
2.23. f(x) = arcsin \sqrt{x}.  2.24. f(x) = arcsin e^{−x}.

3. Sa se calculeze derivata functiei:

3.1. f(x) = ln tg \frac{x}{2}.  3.2. f(x) = ln (x + \sqrt{x^{2} + 1}).

3.3. f(x) = ln \sqrt[4]{\frac{1 − sin x}{1 + sin x}}.  3.4. f(x) = ln \frac{x^{2} − 1}{x^{2} + 1}.

3.5. f(x) = ln sin \frac{2x + 4}{x + 1}.  3.6. f(x) = ln tg (\frac{x}{2} + \frac{π}{4} ).
3.7. f(x) = arctg \sqrt{4x − 1}.  3.8. f(x) = \sqrt{x} − arctg \sqrt{x}.
3.9. f(x) = arctg \frac{1 + x}{1 − x}.  3.10. f(x) = arctg \frac{x}{1 + \sqrt{1 + x^{2}}} .

3.11. f(x) = arcsin \frac{1 − x}{\sqrt{2}}.  3.12. f(x) = arcsin \sqrt{1 − x^{2}}.

3.13. f(x) = arccos \frac{1 − x^{2}}{1 + x^{2}} .  3.14. f(x) = cos (2 arccos x).

3.15. f(x) = e^{\sqrt{\frac{1+x}{1−x}}} .  3.16. f(x) = e^{tg ^{\frac{1}{x}}} .
3.17. f(x) = tg^{2} x + ln cos^{2} x. 3.18. f(x) = arcctg (ctg^{2} x).

3.19. f(x) = \sqrt{2x^{2} + \sqrt{x^{2} + 1}}. 3.20. f(x) = \sqrt{2 + x^{2}}\sqrt[3]{3 + x^{3}}.

4. Sa se calculeze derivata functiei:

4.1. f(x) = ln (2x − 3 + \sqrt{4x^{2} − 12x + 10}) − arctg(2x − 3)\sqrt{4x^{2} − 12x + 10}.
4.2. f(x) = x^{2}\sqrt{x^{4} + 1} + ln (x^{2} + \sqrt{x^{4} + 1}) .
4.3. f(x) = x + e^{−x} arctg e^{x} − ln \sqrt{1 + e^{2x}}.
4.4. f(x) = \sqrt{49x^{2} + 1} arctg 7x − ln (7x + \sqrt{49x^{2} + 1}) .
4.5. f(x) = arcsin e^{−2x} + ln (e^{2x} + \sqrt{e^{4x} − 1}) .
4.6. f(x) = \frac{3 − sin x}{2}\sqrt{cos^{2} x − 2 sin x} + 2 arcsin \frac{1 + sin x}{\sqrt{2}}·

4.7. f(x) = arctg \sqrt{e^{x}} + e^{x} arcsin \sqrt{\frac{e^{x}}{e^{x} + 1}} − \sqrt{e^{x}}.

4.8. f(x) = 2\sqrt{3} arctg \frac{\sqrt{3}}{1 − 2x^{2}} + ln \frac{x^{4} − x^{2} + 1}{x^{4} + 2x^{2} + 1}·

4.9. f(x) = ln \frac{2 (x^{2} + 2x + 2)}{2x^{2} + 2x + 1} + 4 arctg(x + 1) − arctg(2x + 1).

4.10. f(x) = \frac{5x + 2}{x^{2} + x + 1} + ln \sqrt[3]{\frac{(x − 1)^{2}}{x^{2} + x + 1}} + \frac{8}{\sqrt{3}} arctg \frac{2x + 1}{\sqrt{3}}·

4.11. f(x) = x ln (\sqrt{1 − x} + \sqrt{1 + x}) + \frac{1}{2}(arcsin x − x).
4.12. f(x) = (3x − 2)^{4} arcsin \frac{1}{3x − 2} + (3x^{2} − 4x + 2) \sqrt{9x^{2} − 12x + 3}.
4.13. f(x) = e^{2 arcsin x} [cos(2 arcsin x) + sin(2 arcsin x)] .
4.14. f(x) = \sqrt{1 + \sqrt[3]{1 + \sqrt[4]{1 + x^{4}}}}.
4.15. f(x) = \frac{2}{3x − 2}\sqrt{12x − 9x^{2} − 3} + ln \frac{1 + \sqrt{12x − 9x^{2} − 3}}{3x − 2}.
4.16. f(x) = x (2x^{2} + 5) \sqrt{x^{2} + 1} + 3 ln (x + \sqrt{x^{2} + 1}).
4.17. f(x) = \sqrt{x^{2} + 5x + 4} + 3 ln (\sqrt{x + 4} + \sqrt{x + 1}).
4.18. f(x) = \frac{x arcsin 2x}{\sqrt{1 − 4x^{2}}} + ln \sqrt{1 − 4x^{2}}.

4.19. f(x) = \frac{1}{4\sqrt{3}} ln \frac{\sqrt{x^{2} + 2} − x\sqrt{3}}{\sqrt{x^{2} + 2} + x\sqrt{3}} + \frac{1}{2} arctg \frac{\sqrt{x^{2} + 2}}{x}.

4.20. f(x) = \frac{cos x}{3(2 + sin x)} + \frac{4}{3\sqrt{3}} arctg \frac{2 tg \frac{x}{2} + 1}{\sqrt{3}}.

4.21. f(x) = \frac{1}{cos x} + \frac{1}{3 cos^{3} x} − \frac{1}{2} ln \frac{1 + cos x}{1 − cos x}.
4.22. f(x) = 2\sqrt{1 − x^{2}} arcsin x − 2x + x(arcsin x)^{2}.
4.23. f(x) = \frac{ln(1 + sin x)}{tg x} + x − ln tg \frac{x}{2}.

4.24. f(x) = log_{\frac{1}{2}} (x + \frac{1}{2})^{2} + log_{2} \sqrt{4x^{2} + 4x + 1}.
4.25. f(x) = x^{x}.
4.26. f(x) = sin x^{cos x}.
4.27. f(x) = x + x^{x} + x^{x^{x}}.
4.28. f(x) = x^{e^{x}}.
4.29. f(x) = x^{e^{sin x}}.
4.30. f(x) = x^{3^{x}}2^{x}.

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5. Sa se studieze derivabilitatea urmatoarelor functii:

5.1. f : ℝ ⟶ ℝ,  f(x) = |x^{3} − 4x|.
5.2. f : (−\frac{1}{3}, ∞) ⟶ ℝ, f(x) = \{ \begin{matrix}ln(1 + 3x), & dacă & −\frac{1}{3} < x ≤ 0 \\ 3x, & dacă & x > 0. \end{matrix}
5.3. f : ℝ ⟶ ℝ,  f(x) = \{ \begin{matrix}sin^{3} x sgn x, & dacă & |x| ≤ \frac{π}{4} \\ \frac{3\sqrt{2}}{4}x sgn x − \frac{\sqrt{2}(3π − 4)}{4}, & dacă & |x| > \frac{π}{4}. \end{matrix}
5.4. f : ℝ ⟶ ℝ,  f(x) = \{ \begin{matrix}tg (x^{3} + x^{2} sin \frac{2}{x}), & x ≠ 0 \\ 0, & x = 0. \end{matrix}
5.5. f : ℝ ⟶ ℝ,  f(x) = \{ \begin{matrix}\sqrt[3]{1 − 2x^{3} sin \frac{5}{x}} − 1 + x, & x ≠ 0 \\ 0, & x = 0. \end{matrix}
5.6. f : ℝ ⟶ ℝ,  f(x) = \frac{|x + 1| − |4 − x|}{|x| + |x − 5|}.
5.7. f : ℝ ⟶ ℝ,  f(x) = | cos x|.
5.8. f : ℝ ⟶ ℝ,  f(x) = \{ \begin{matrix}x, & dacă & x ∈ ℚ \\ 0, & dacă & x ∈ ℝ\ℚ. \end{matrix}
5.9. f : ℝ ⟶ ℝ,  f(x) = \{ \begin{matrix}arctg ax, & dacă & |x| ≤ 1, a ∈ ℝ \\ b sgn x + \frac{x − 1}{2}, & dacă & |x| > 1, b ∈ ℝ. \end{matrix}
5.10. f : ℝ ⟶ ℝ,  f(x) = \{ \begin{matrix}2^{\frac{1}{x−1}} , & dacă & x < 1 \\ 0, & dacă & x = 1 \\ ln(x^{2} − 2x + 2), & dacă & x > 1. \end{matrix}

6. Sa se calculeze derivatele de ordinul n (n ∈ ℤ, n ≥ 1) ale functiilor urmatoare:

6.1. f(x) = xe^{2x}.  6.2. f(x) = \frac{12x − 1}{6x − 1}.
6.3. f(x) = sin 3x + cos (x + 2).  6.4. f(x) = ln (x + 3).
6.5. f(x) = (x − 1)^{n}(x − 2)^{n}.  6.6. f(x) = x^{n}e^{−x}.
6.7. f(x) = sin x.  6.8. f(x) = cos x.

6.9. f(x) = sin^{2} x.  6.10. f(x) = sin^{4} x + cos^{4} x.
6.11. f(x) = \frac{x}{x^{2} − 4x − 12}.  6.12. f(x) = \frac{3}{x^{2} − x − 2}.
6.13. f(x) = x sin x.  6.14. f(x) = arctg x.
6.15. f(x) = \frac{1}{\sqrt{x − 1}}.  6.16. f(x) = e^{x} sin x.

6.17. f(x) = e^{x} cos 2x.  6.18. f(x) = \frac{ln x}{x}.
6.19. f(x) = \frac{2x + 1}{3x + 2}.  6.20. f(x) = \sqrt[3]{e^{2x−1}}.

7. Utilizand diferentiale, sa se calculeze cu aproximatie:

7.1. f(x) = x^{5} , x = 3, 01.  7.2. f(x) = x^{6} , x = 1, 997.
7.3. f(x) = \sqrt[3]{x^{2}} , x = 1, 029.  7.4. f(x) = \sqrt{3 + x + cos x} , x = 0, 01.
7.5. f(x) = \sqrt{\frac{3 − x}{1 + x}} , x = −0, 85.  7.6. f(x) = \frac{1}{\sqrt{3x + 2}} , x = 0, 668.
7.7. f(x) = arcsin x , x = 0, 08.  7.8. f(x) = arctg x , x = 1, 03.
7.9. f(x) = sin x , x = 31^{∘}.  7.10. f(x) = ln tg x , x = 48^{∘}.
7.11. f(x) = \frac{x + \sqrt{10 − x^{2}}}{2} , x = 0, 99.  7.12. f(x) = \sqrt{x^{2} + 12} , x = 1, 98.

8. Sa se calculeze derivata y′_{x}:

8.1. \{ \begin{matrix}x = sin^{2}t, \\ y = cos^{2}t. \end{matrix}  8.2. \{ \begin{matrix}x = e^{−t}, \\ y = t^{2}. \end{matrix}

8.3. \{ \begin{matrix}x = \sqrt{t}, \\ y = \sqrt[3]{t}. \end{matrix}  8.4. \{ \begin{matrix}x = e^{t}, \\ y = arcsin t. \end{matrix}

8.5. \{ \begin{matrix}x = \frac{3at}{1 + t^{3}} , \\ y = \frac{3at^{2}}{1 + t^{3}} . \end{matrix}  8.6. \{ \begin{matrix}x = \frac{1}{t + 1}, \\ y = \frac{t}{t + 1}. \end{matrix}

8.7. \{ \begin{matrix}x = arctg e^{t}, \\ y = \sqrt{e^{2t} + 1}. \end{matrix}  8.8. \{ \begin{matrix}x = arctg t, \\ y = ln \frac{1 + t^{2}}{t + 1} . \end{matrix}

8.9. \{ \begin{matrix}x = \frac{t}{1 − t^{2}} arcsin t + ln \sqrt{1 − t^{2}}, \\ y = \frac{t}{\sqrt{1 − t^{2}}} . \end{matrix}  8.10. \{ \begin{matrix}x = ln tg t, \\ y = cosec^{2}t. \end{matrix}

8.11. \{ \begin{matrix}x = \frac{5t^{2} + 2}{5t^{3}}, \\ y = sin (\frac{1}{3}t^{3} + t). \end{matrix}  8.12. \{ \begin{matrix}x = \sqrt{4 − t^{2}}, \\ y = tg \sqrt{2 + t}. \end{matrix}

8.13. \{ \begin{matrix}x = e^{t} cos t, \\ y = e^{t} sin t. \end{matrix}  8.14. \{ \begin{matrix}x = a(sin t − t cos t), \\ y = a(cos t + t sin t). \end{matrix}

8.15. xy + ln y = 1.  8.16. \sqrt{x} + \sqrt{y} = 1.
8.17. \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1.  8.18. e^{y} + xy = 2e.
8.19. x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}} .  8.20. y^{5} + y^{3} + y − x = 0.
8.21. arctg \frac{y}{x} = ln \sqrt{x^{2} + y^{2}}.  8.22. y^{2} = 2px.
8.23. x^{2} + y^{2} − 6x + 10y − 2 = 0.  8.24. x^{2}y + arctg \frac{y}{x} = 0.

9. Sa se scrie ecuatiile tangentelor la graficele functiilor in punctele specificate:

9.1. f(x) = x^{2} − x − 12 , x = 3.  9.2. f(x) = \frac{1}{3}(3x − x^{3}) , x = 2.

9.3. f(x) = \frac{x^{3} + 1}{x^{3} − 1} , x = 0.  9.4. f(x) = \frac{x}{x^{2} + 1} , x = −1.

9.5. f(x) = \frac{x^{2} − x − 2}{x^{2} − 3x} , x = 2.  9.6. f(x) = ln \frac{x^{2} − 2x + 1}{x^{2} + x + e} , x = 0.
9.7. f(x) = cos 2x − 2 sin x , x = \frac{π}{2} .  9.8. f(x) = arctg \frac{1}{x} , x = 1.
9.9. f(x) = \frac{x}{\sqrt[3]{x + 1}} , x = −2.  9.10. f(x) = 4 tg x − \frac{sin x}{cos^{2}x} , x = \frac{π}{4} .

10. Sa se determine in ce puncte si sub ce unghi se intersecteaza graficele functiilor:

10.1. f_{1} (x) = sin x, f_{2} (x) = \sqrt{3} cos x.  10.2. f_{1} (x) = x^{2}, f_{2} (x) = x.
10.3. f_{1} (x) = x^{3}, f_{2} (x) = x^{2}.  10.4. f_{1} (x) = (x − 2)^{2} , f_{2} (x) = 4 − x^{2}.
10.5. f_{1} (x) = \frac{1}{\sqrt[3]{x}}, f_{2} (x) = x.  10.6. f_{1} (x) = \frac{1}{x^{3}} , f_{2} (x) = x^{2}.
10.7. f_{1} (x) = 4x^{2} + 2x − 8, f_{2} (x) = x^{3} − x + 10. 10.8. f_{1} (x) = ln x, f_{2} (x) = 2 − \frac{x}{e}.
10.9. f_{1} (x) = 3x − x^{2}, f_{2} (x) = x^{2} − x.  10.10. f_{1} (x) = sin x, f_{2} (x) = cos x.

11. Sa se studieze monotonia si sa se determine punctele de extrem pentru fiecare din functiile f pe domeniul lor maxim de definitie:

11.1. f (x) = x^{2} − x − 12.  11.2. f (x) = 6x − x^{2}.
11.3. f (x) = 3x^{3} − 4x^{2} + 1.  11.4. f (x) = x^{3} − 6x^{2} + 2.
11.5. f (x) = (x + 1)^{2} (x − 4)^{3} .  11.6. f (x) = x^{2} − 8 ln x.
11.7. f (x) = \sqrt[3]{(2 − x) (1 − x)^{2}}.  11.8. f (x) = (x − 1) \sqrt{x^{2} − 1}.
11.9. f (x) = ln (1 + x) − x + \frac{x^{2}}{2} .  11.10. f (x) = \frac{x^{2}}{x − 1}.
11.11. f (x) = ln \sqrt{1 + x^{2}} + arctg x.  11.12. f (x) = x^{2}e^{\frac{1}{x}} .
11.13. f (x) = ln (4x^{2} + 1) − 8 arctg 2x.  11.14. f (x) = \frac{x^{3}}{3} e^{−x}.
11.15. f (x) = ln x + arctg x.  10.16 f (x) = x^{2} ln x.
11.17. f (x) = x − 2 arctg (x − 1) − 1.  11.18. f (x) = sin^{3} x + cos^{3} x.
11.19. f (x) = cos x + \frac{1}{2} sin 2x.  11.20. f (x) = \frac{1}{x − 1} + \frac{2 (x − 1)}{x^{2} − 2x}.

12. Sa se determine intervalele de concavitate, convexitate si eventualele puncte de inflexiune pentru functiile urmatoare:

12.1. f(x) = 2x^{4} − 3x^{2} + 3x − 2.  12.2. f(x) = x^{4} + 4x^{3}.
12.3. f(x) = 3x^{2} − x^{3} + 1.  12.4. f(x) = x + cos x.
12.5. f(x) = e^{−x^{2}} + 2x.  12.6. f(x) = ln (1 + x^{2}).

12.7. f(x) = \frac{(x + 1)^{2}}{x^{3}} .  12.8. f(x) = \begin{matrix}ln & (x + 1) \\ √ & \overline{x + 1} \end{matrix}.

12.9. f(x) = (\frac{x}{2 − x})^{4} .  12.10. f(x) = sin x + \frac{1}{3} sin 3x.

12.11. f(x) = \sqrt[3]{x − 1} − \sqrt[3]{x}.  12.12. f(x) = sin x − sin^{3} x.
12.13. f(x) = sin^{4} x − cos^{4} x.  12.14. f(x) = x^{5} − 10x^{2} + 7x.
12.15. f(x) = tgx + cos x.  12.16. f(x) = x + ln x^{2}.

12.17. f(x) = ln \frac{x}{x − 3}.  12.18. f(x) = \frac{\sqrt{x + 1}}{x}.

12.19. f(x) = e^{x} − \frac{1}{2}x^{2} + 1.  12.20. f(x) = 3x + 2 sin \frac{x}{2}.

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13. Sa se reprezinte grafic urmatoarele functii, f : D ⟶ ℝ, D – fiind domeniul maxim de

definitie:

13.1. f(x) = 3x − x^{3}.  13.2. f(x) = 2 − 3x − x^{3}.
13.3. f(x) = \frac{1}{16}x^{2}(x − 4)^{2}.  13.4. f(x) = x^{2} − x^{4}.
13.5. f(x) = x(2x^{2} + 9x + 12).  13.6. f(x) = (x − 1)^{2}(3 − x)^{2}.
13.7. f(x) = \frac{3x − 2}{x^{3}}.  13.8. f(x) = \frac{x^{3} + 4}{x^{2}}.
13.9. f(x) = (\frac{x}{x − 1})^{2} ·  13.10. f(x) = \frac{3x^{4} + 1}{x^{3}}·
13.11. f(x) = \frac{x^{3}}{x − 1}·  13.12. f(x) = 3x + \frac{6}{x} − \frac{1}{x^{3}} ·
13.13. f(x) = \frac{3}{x + 2} − \frac{3}{x − 2} − 1.  13.14. f(x) = \begin{matrix}ln & (x + 1) \\ √ & \overline{x + 1} \end{matrix}·
13.15. f(x) = sin x − sin^{2} x.  13.16. f(x) = cos 3x + 3 cos x.
13.17. f(x) = sin x + \frac{1}{2} sin 2x.  13.18. f(x) = cos x cos 3x.
13.19. f(x) = arccos \frac{2x}{1 + x^{2}} ·  13.20. f(x) = arcsin \frac{1 − x^{2}}{1 + x^{2}} ·
13.21. f(x) = ln x − x + 1.  13.22. f(x) = x^{2} ln x.
13.23. f(x) = \frac{ln x}{x}·  13.24. f(x) = ln (\frac{x − 5}{x}) + 2.
13.25. f(x) = x arctg x.  13.26. f(x) = arctg sin x.
13.27. f(x) = ln(sin x − cos x).  13.28. f(x) = x^{\frac{2}{3}} e^{− \frac{x^{2}}{3}} .
13.29. f(x) = \frac{e^{x+2}}{x + 2}·  13.30. f(x) = e^{sin x+cos x}.
13.31. f(x) = \sqrt[3]{x(x^{2} − 1)}.  13.32. f(x) = \sqrt[3]{(x − 2)(x + 1)^{2}}.

14. Sa se calculeze limitele urmatoare folosind regula lui l’Hospital:

14.1. lim_{x→1} \frac{x^{3} − 5x^{2} + 4}{2x^{3} − x^{2} − 1}·  14.2. lim_{x→1} \frac{x^{5} − 1}{ln x}·
14.3. lim_{x→0} \frac{sin 5x}{2x}·  14.4. lim_{x→ \frac{π}{2}} \frac{cos 5πx}{cos 3πx}·

14.5. lim_{x→0} \frac{tg x − x}{sin x − x}·  14.6. lim_{x→0} \frac{e^{x} − e^{−x}}{ln(1 + x)}·

14.7. lim_{x→∞} \frac{ln(1 + \frac{1}{x^{2}} )}{π − 2 arctg x}·  14.8. lim_{x→0} \frac{ln cos 2x}{ln cos 3x}·

14.9. lim_{x→0} \frac{sin 2x − 2xe^{x} + 3x^{2}}{arctg x − sin x − \frac{x^{3}}{6}} ·  14.10. lim_{x→ \frac{π}{4}} \frac{ln tg x}{ctg 2x}·

14.11. lim_{x→∞} \frac{π − 2 arctg x}{e^{\frac{2}{x}} − 1}·  14.12. lim_{x→1} \frac{ln x − x + 1}{tg^{2}(x − 1)}·

14.13. lim_{x→∞} \frac{x^{2}}{e^{x}} ·  14.14. lim_{x→∞} \frac{x^{4}}{e^{x}} ·
14.15. lim_{x→1+} \frac{ln(x − 1)}{ctg πx}·  14.16. lim_{x→0+} \frac{ln x}{ln sin x}·
14.17. lim_{x→0} x ctg πx.  14.18. lim_{x→ \frac{π}{2}} (x − \frac{π}{2} ) tg x.
14.19. lim_{x→0}(ctg x arcsin x).  14.20. lim_{x→0} sin x ln(ctg x).
14.21. lim_{x→2}(x − 2) tg \frac{πx}{4}·  14.22. lim_{x→3}(x − 3) ctg \frac{πx}{3}·

14.23. lim_{x→1} (\frac{1}{x − 1} − \frac{1}{ln x})·  14.24. lim_{x→0} (\frac{1}{x^{2}} − ctg^{2} x).

14.25. lim_{x→3} (\frac{2x − 3}{x^{2} − 7x + 12} − \frac{1}{(x − 2) ln(x − 2)}). 14.26. lim_{x→0} (\frac{1}{x} − \frac{1}{arcsin x}).

14.27. lim_{x→1} (\frac{2}{1 − x^{2}} − \frac{3}{1 − x^{3}} ).  14.28. lim_{x→0} (\frac{1}{x} − \frac{1}{e^{2x} − 1}).

14.29. lim_{x→0}(cos x)^{\frac{1}{x^{2}}} .  14.30. lim_{x→∞} (\frac{2}{π} arctg x)^{x}.
14.31. lim_{x→0}(x + 3^{x})^{\frac{2}{x}} .  14.32. lim_{x→0}(x + e^{x})^{\frac{1}{x}} .
14.33. lim_{x→π+}(x − π)^{sin x}.  14.34. lim_{x→0+} | ln x|^{\frac{x}{2}} .

14.35. lim_{x→0+} (\frac{1}{x})^{sin x}.  14.36. lim_{x→0} (\frac{sin x}{x})^{\frac{1}{x^{2}}} ·