Задачник Кузнецова Пределы Задачи 17-20

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Содержание

1 Задачи 17
2 Задачи 18
3 Задачи 19
4 Задачи 20

Задачи 17

Вычислить предел функции:

Задача	Условие	Задача	Условие
17-1	$\lim_{x\to 0} \left(\frac{\sin {2x}}{x}\right)^{1+x}$	17-2	$\lim_{x\to 0} \left(\frac{2+x}{3-x}\right)^x$
17-3	$\lim_{x\to 0} \left(\frac{\sin {4x}}{x}\right)^{\frac{2}{x+2}}$	17-4	$\lim_{x\to 0} \left(\frac{e^{3x}-1}{x}\right)^{\cos^{2}\left(\frac{\pi}{4}+x \right)}$
17-5	$\lim_{x\to 0} \left(\cos {x}\right)^{x+3}$	17-6	$\lim_{x\to 0} \left(\frac{x^2+4}{x+2} \right)^{x^2+3}$
17-7	$\lim_{x\to 0} \left(\frac{\ln {\left(1+x \right)}}{6x} \right)^{\frac{x}{x+2}}$	17-8	$\lim_{x\to 0} \left(\frac{\operatorname{tg}{4x}}{x}\right)^{2+x}$
17-9	$\lim_{x\to 0} \left(\frac{e^{x^3}-1}{x^2}\right)^{\frac{8x+3}{1+x}}$	17-10	$\lim_{x\to 0} \left(\frac{x+2}{x+4}\right)^{\cos {x}}$
17-11	$\lim_{x\to 0} \left(\frac{\sin{6x}}{2x}\right)^{2+x}$	17-12	$\lim_{x\to 0} \left(\frac{e^{x^2}-1}{x^2} \right)^{\frac{6}{1+x}}$
17-13	$\lim_{x\to 0} \left(\frac{\sin{2x}}{\sin {3x}}\right)^{x^2}$	17-14	$\lim_{x\to 0} \left(\operatorname{tg} {\left(x+\frac{\pi }{3} \right)}\right)^{x+2}$
17-15	$\lim_{x\to 0} \left(\frac{x^3+8}{3x^2+10}\right)^{x+2}$	17-16	$\lim_{x\to 0} \left(\sin {(x+2)}\right)^{\frac{3}{3+x}}$
17-17	$\lim_{x\to 0} \left(\frac{2^{2x}-1}{x} \right)^{x+1}$	17-18	$\lim_{x\to 0} \left(\frac{x^4+5}{x+10}\right)^{\frac {4}{x+2}}$
17-19	$\lim_{x\to 0} \left(\frac{11x+8}{12x+1} \right)^{\cos^{2}{x}}$	17-20	$\lim_{x\to 0} \left(\frac{x^3+1}{x^3+8} \right)^{\frac{2}{x+1}}$
17-21	$\lim_{x\to 0} \left(\frac{\ln {\left(1+x^2 \right)}}{x^2} \right)^{\frac{3}{x+8}}$	17-22	$\lim_{x\to 0} \left(\cos {\left(\frac{x}{\pi }\right)} \right)^{1+x}$
17-23	$\lim_{x\to 0} \left(\frac{\arcsin {x}}{x} \right)^{\frac{2}{x+5}}$	17-24	$\lim_{x\to 0} \left(\frac{\operatorname{arctg}{3x}}{x} \right)^{x+2}$
17-25	$\lim_{x\to 0} \left(e^x+x\right)^{\cos {x^4}}$	17-26	$\lim_{x\to 0} \left(\frac{\sin {5x^2}}{\sin {x}} \right)^{\frac{1}{x+6}}$
17-27	$\lim_{x\to 0} \left(\operatorname{tg}{\left(\frac{\pi}{4}-x\right)} \right)^{\left(e^x-1\right)/x}$	17-28	$\lim_{x\to 0} \left(6-\frac{5}{\cos {x}} \right)^{\operatorname{tg}^{2}{x}}$
17-29	$\lim_{x\to 0} \left(\frac{1+8x}{2+11x} \right)^{\frac{1}{x^2+1}}$	17-30	$\lim_{x\to 0} \left(\frac{\arcsin^{2}{x}}{\arcsin^{2}{4x}} \right)^{2x+1}$
17-31	$\lim_{x\to 0} \left(\frac{x^3+4}{x^3+9} \right)^{\frac{1}{x+2}}$

Задачи 18

Вычислить предел функции:

Задача	Условие	Задача	Условие
18-1	$\lim_{x\to 1} \left(\frac{3x-1}{x+1}\right)^{\frac{1}{\sqrt[3]{x}-1}}$	18-2	$\lim_{x\to a} \left(\frac{\sin {x}}{\sin {a}} \right)^{\frac{1}{x-a}}$
18-3	$\lim_{x\to 1} \left(\frac{2x-1}{x} \right)^{\frac{1}{\sqrt[3]{x}-1}}$	18-4	$\lim_{x\to 2} \left(\frac{\cos {x}}{\cos {2}} \right)^{\frac{1}{x-2}}$
18-5	$\lim_{x\to 8} \left(\frac{2x-7}{x+1} \right)^{\frac{1}{\sqrt[3]{x}-2}}$	18-6	$\lim_{x\to \frac{\pi}{4}} \left(\operatorname{tg}{x}\right)^{1/\cos{\left(\frac{3\pi}{4}-x\right)}}$
18-7	$\lim_{x\to 1} \left(\frac{2x-1}{x} \right)^{1/ \left(\sqrt[5]{x}-1\right)}$	18-8	$\lim_{x\to a} \left(2-\frac{x}{a} \right)^{\operatorname{tg}{\left(\frac{\pi x}{2a}\right)}}$
18-9	$\lim_{x\to 2\pi} \left(\cos {x} \right)^{\frac{\operatorname{ctg}{2x}}{\sin {3x}}}$	18-10	$\lim_{x\to 2\pi} \left(\cos {x} \right)^{\frac{1}{\sin^{2}{2x}}}$
18-11	$\lim_{x\to 3} \left(\frac{6-x}{3} \right)^{\operatorname{tg}{\left(\frac{\pi x}{6}\right)}}$	18-12	$\lim_{x\to 4\pi} \left(\cos {x} \right)^{\frac{\operatorname{ctg}{x}}{\sin {4x}}}$
18-13	$\lim_{x\to 1} \left(3-2x\right)^{\operatorname{tg}{\left(\frac{\pi x}{2}\right)}}$	18-14	$\lim_{x\to 4\pi} \left(\cos {x}\right)^{\frac{5}{\operatorname{tg}{5x}\sin {2x}}}$
18-15	$\lim_{x\to 3} \left(\frac{9-2x}{3} \right)^{\operatorname{tg}{\left(\frac{\pi x}{6}\right)}}$	18-16	$\lim_{x\to \frac{\pi}{2}} \left(\sin {x}\right)^{6\operatorname{tg}{x}\cdot \operatorname{tg}{3x}}$
18-17	$\lim_{x\to 1} \left(2e^{x-1}-1\right)^{\frac{x}{x-1}}$	18-18	$\lim_{x\to \frac{\pi}{2}} \left(\operatorname{tg}{\left(\frac{x}{2}\right)}\right)^{1 / \left(x-\frac{\pi}{2}\right)}$
18-19	$\lim_{x\to 1} \left(2e^{x-1}-1\right)^{\frac{3x-1}{x-1}}$	18-20	$\lim_{x\to \frac{\pi}{2}} \left(1+\cos {3x} \right)^{\sec {x}}$
18-21	$\lim_{x\to 2} \left(2e^{x-2}-1\right)^{\frac{3x+2}{x-2}}$	18-22	$\lim_{x\to 1} \left(\frac{\sin {(x-1)}}{x-1}\right)^{\frac{\sin {(x-1)}}{x-1-\sin {(x-1)}}}$
18-23	$\lim_{x\to 1} \left(\frac{2-x}{x} \right)^{\frac{1}{\ln {(2-x)}}}$	18-24	$\lim_{x\to \frac{\pi}{2}} \left(\operatorname{ctg}{\left(\frac{x}{2}\right)}\right)^{\frac{1}{\cos {x}}}$
18-25	$\lim_{x\to 1} \left(2-x\right)^{\sin {\left(\frac{\pi x}{2}\right)} / \ln {(2-x)}}$	18-26	$\lim_{x\to 3} \left(\frac{\sin{x}}{\sin {3}} \right)^{\frac{1}{x-3}}$
18-27	$\lim_{x\to 1} \left(\frac{x+1}{2x} \right)^{\frac{\ln {(x+2)}}{\ln {(2-x)}}}$	18-28	$\lim_{x\to \frac{\pi}{2}} \left(\sin {x} \right)^{\frac{18\sin{x}}{\operatorname{ctg}{x}}}$
18-29	$\lim_{x\to 1} \left(\frac{1}{x}\right)^{\frac{\ln {(x+1)}}{\ln {(2-x)}}}$	18-30	$\lim_{x\to \pi} \left(\operatorname{ctg}{\left(\frac{x}{4}\right)} \right)^{1 / \cos{\left(\frac{x}{2}\right)}}$
18-31	$\lim_{x\to 1} \left(\frac{2x-1}{x}\right)^{\frac{\ln {(3+2x)}}{\ln {(2-x)}}}$

Задачи 19

Вычислить предел функции:

Задача	Условие	Задача	Условие
19-1	$\lim_{x\to e} \left(\frac{\ln {x}-1}{x-e} \right)^{\sin {\left(\frac{\pi}{2e}x\right)}}$	19-2	$\lim_{x\to \frac{\pi}{4}} \left(\operatorname{tg}{x} \right)^{\operatorname{ctg}{x}}$
19-3	$\lim_{x\to \frac{\pi}{4}} \left( \frac{\ln {\operatorname{tg}{x}}}{1-\operatorname{ctg}{x}} \right)^{1 / \left(x+\frac{\pi}{4}\right)}$	19-4	$\lim_{x\to 2} \left(\sin {x} \right)^{\frac{3}{1+x}}$
19-5	$\lim_{x\to 2} \left(\frac{\sin{(3\pi x)}}{\sin {(\pi x)}} \right)^{\sin^{2}{(x-2)}}$	19-6	$\lim_{x\to \frac{\pi}{6}} \left(\sin {x}\right)^{\frac{6x}{\pi}}$
19-7	$\lim_{x\to 3} \left(2-\frac{x}{3} \right)^{\sin {\left(\pi x\right)}}$	19-8	$\lim_{x\to 1} \left(\frac{1+x}{2+x}\right)^{\frac{1-x^2}{1-x}}$
19-9	$\lim_{x\to 1} \left(1+e^x\right)^{\frac{\sin {\pi x}}{1-x}}$	19-10	$\lim_{x\to 1} \left(\frac{\operatorname{tg}{9\pi x}}{\sin {4\pi x}} \right)^{\frac{x}{x+1}}$
19-11	$\lim_{x\to 3} \left(\frac{\arcsin {\left(x-3\right)}}{\sin {3\pi x}} \right)^{x^2-8}$	19-12	$\lim_{x\to \frac{\pi}{4}} \left(\sin {2x} \right)^{\left(x^2-\frac{\pi^2}{16}\right) / \left(x-\frac{\pi}{4}\right)}$
19-13	$\lim_{x\to 1} \left(\operatorname{arctg}{\left(\frac{x-\frac{3}{4}}{(x-1)^2} \right)}\right)^{x+1}$	19-14	$\lim_{x\to \pi } \left(\operatorname{ctg}{\frac{x}{4}} \right)^{\sin {(x-\pi)}}$
19-15	$\lim_{x\to a} \left(\frac{\sin{x}-\sin {a}}{x-a}\right)^{\frac{x^2}{a^2}}$	19-16	$\lim_{x\to 2} \left(\frac{\sqrt {x+2} -2}{x^2-4}\right)^{\frac{1}{x}}$
19-17	$\lim_{x\to \frac{\pi}{4}} \left(\sin {x}+\cos {x}\right)^{\frac{1}{\operatorname{tg}{x}}}$	19-18	$\lim_{x\to \frac{\pi}{8}} \left(\operatorname{tg}{2x}\right)^{\sin {\left(\frac{\pi}{8}+x\right)}}$
19-19	$\lim_{x\to 1} \left(\arcsin {x}\right)^{\operatorname{tg}{\pi x}}$	19-20	$\lim_{x\to \pi} \left(x+\sin {x} \right)^{\sin {x}+x}$
19-21	$\lim_{x\to 1} \left(\ln^{2}{ex} \right)^{\frac{1}{x^2+1}}$	19-22	$\lim_{x\to 1} \left(\sqrt{x} +1\right)^{\frac{\pi}{\operatorname{arctg}{x}}}$
19-23	$\lim_{x\to 1} \left(\frac{x^3-1}{x-1} \right)^{\frac{1}{x^2}}$	19-24	$\lim_{x\to 1} \left(\frac{e^{\sin {\pi x}-1}}{x-1} \right)^{x^2+1}$
19-25	$\lim_{x\to 2} \left(\cos {\pi x}\right)^{\operatorname{tg}{(x-2)}}$	19-26	$\lim_{x\to \frac{1}{2}} \left(\arcsin {x}+\arccos {x}\right)^{\frac{1}{x}}$
19-27	$\lim_{x\to \frac{\pi}{2}} \left(\cos {x}+1 \right)^{\sin {x}}$	19-28	$\lim_{x\to 1} \left(\sqrt[3]{x}+x-1 \right)^{\sin {\left(\frac{\pi x}{4}\right)}}$
19-29	$\lim_{x\to 1} \left(\frac{x^2+2x-3}{x^2+4x-5}\right)^{\frac{1}{2-x}}$	19-30	$\lim_{x\to 1} \left(\frac{1+\cos {\pi x}}{\operatorname{tg}^{2}{\pi x}} \right)^{x^2}$
19-31	$\lim_{x\to 1} \left(\frac{e^{2x}-e^2}{x-1} \right)^{x+1}$

Задачи 20

Вычислить предел функции или числовой последовательности:

Задача	Условие	Задача	Условие
20-1	$\lim_{x\to 0} \sqrt {4\cos{3x}+x\cdot\operatorname{arctg}{\left(\frac{1}{x}\right)}}$	20-2	$\lim_{x\to \frac{\pi}{2}} \sqrt {3\sin {x}+\left(2x-\pi\right)\sin {\frac{x}{2x-\pi }}}$
20-3	$\lim_{n\to \infty} \frac{2n-\sin{n}}{\sqrt {n} -\sqrt[3]{n^3-7}}$	20-4	$\lim_{x\to 0} \frac{\operatorname{tg}{x}\cdot \cos {\left(\frac{1}{x}\right)}+\lg {(2+x)}}{\lg {(4+x)}}$
20-5	$\lim_{n\to \infty} \frac{e^{1 / n}+\sin \left(\frac{n}{n^2+1}\right)\cdot \cos {n}}{1+\cos {\left(\frac{1}{n}\right)}}$	20-6	$\lim_{n\to \infty} \frac{\sqrt[4]{2+n^5}-\sqrt {2n^3+3}}{\left(n+\sin {n} \right)\sqrt {7n} }$
20-7	$\lim_{x\to \frac{\pi}{4}} \frac{\sqrt[3]{\operatorname{tg}{x}}+\left(4x-\pi \right)\cos {\left(\frac{x}{4x-\pi }\right)}}{\lg {\left(2+\operatorname{tg}{x}\right)}}$	20-8	$\lim_{n\to \infty} \left(\sin {\sqrt {n^2+1} \cdot \operatorname{arctg}\frac{n}{n^2+1}} \right)$
20-9	$\lim_{n\to \infty} \frac{n^2-\sqrt{3n^5-7}}{\left(n^2-n\cos {n}+1\right)\sqrt {n}}$	20-10	$\lim_{n\to \infty} \frac{3\sin {n}+\sqrt {n-1}}{n+\sqrt{n+1}}$
20-11	$\lim_{n\to \infty } \frac{\left(1-\cos {n}\right)\sqrt[3]{n}}{\sqrt {2n+1} -1}$	20-12	$\lim_{x\to 0} \ln {\left(2+\sqrt{\operatorname{arctg}{x}\cdot \sin {\left(\frac{1}{x}\right)}} \right)}$
20-13	$\lim_{x\to -2} \sqrt{\frac{1+\cos{\pi x}}{4+\left(x+2 \right)\sin {\left(\frac{x}{x+2}\right)}}}$	20-14	$\lim_{n\to \infty} \frac{n}{\sqrt[3]{n^4-3}+\sin {n}}$
20-15	$\lim_{n\to \infty} \frac{\sqrt[3]{n^2+\cos {n}}+\sqrt{3n^2+2}}{\sqrt[5]{n^6+1}}$	20-16	$\lim_{x\to 0} \frac{\sqrt[3]{\operatorname{tg}{x}}\cdot \operatorname{arctg}{\left(\frac{1}{x}\right)}+3}{2-\lg {\left(1+\sin {x}\right)}}$
20-17	$\lim_{x\to 0} \sqrt {\operatorname{arctg}{x}\cdot \sin^{2}{\left(\frac{1}{x}\right)}+5\cos {x}}$	20-18	$\lim_{x\to 0} \sqrt{4\cos{x}+\sin{\left(\frac{1}{x}\right)}\cdot \ln {\left(1+x\right)}}$
20-19	$\lim_{x\to 0} \sqrt{2\cos^{2}{x}+\left(e^x-1\right)\sin {\left(\frac{1}{x}\right)}}$	20-20	$\lim_{x\to 0} \frac{2+\ln{\left(e+x\sin {\left(\frac{1}{x}\right)}\right)}}{\cos {x}+\sin {x}}$
20-21	$\lim_{x\to 0} \ln {\left(\left(e^{x^2}-\cos {x}\right)\cos {\left(\frac{1}{x}\right)}+\operatorname{tg}{\left(x+\frac{\pi}{3}\right)}\right)}$	20-22	$\lim_{x\to 0} \frac{\cos{x}+\ln{\left(1+x \right)}\sqrt{2+\cos{\left(\frac{1}{x}\right)}}}{2+e^x}$
20-23	$\lim_{x\to 1} \frac{\cos{(2\pi x)}}{2+\left(e^{\sqrt{x-1}}-1 \right)\operatorname{arctg}{\frac{x+2}{x-1}}}$	20-24	$\lim_{x\to 0} \sqrt{\left(e^{\sin{x}}-1\right)\cos{\left(\frac{1}{x}\right)}+4\cos{x}}$
20-25	$\lim_{x\to 0} \frac{\cos{(1+x)}}{\left(2+\sin{\left(\frac{1}{x}\right)} \right)\ln {(1+x)}+2}$	20-26	$\lim_{x\to 2} \sqrt[3]{\lg{(x+2)}+\sin {\sqrt {4-x^2}} \cos{\frac{x+2}{x-2}}}$
20-27	$\lim_{x\to \frac{\pi}{2}} \frac{2+\cos{x}\cdot \sin{\frac{2}{2x-\pi}}}{3+2x\sin{x}}$	20-28	$\lim_{x\to 1} \operatorname{tg}{\left(\cos{x}+\sin{\frac{x-1}{x+1}}\cdot \cos{\frac{x+1}{x-1}}\right)}$
20-29	$\lim_{x\to 0} \sqrt{x\left(2+\sin{\left(\frac{1}{x}\right)} \right)+4\cos{x}}$	20-30	$\lim_{x\to 1} \frac{\sin{x}+\sin{(\pi x)}\cdot \operatorname{arctg}{\frac{1+x}{1-x}}}{1+\cos{x}}$
20-31	$\lim_{n\to \infty} \frac{\sqrt{n^2+3n-1}+\sqrt[3]{2n^2+1}}{n+2\sin{n}}$