Задачник Кузнецова Пределы Задачи 13-16

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Задачи 13

Вычислить предел функции:

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

Задачи 14

Вычислить предел функции:

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

Задачи 15

Вычислить предел функции:

Задача Условие Задача Условие
15-1 \lim_{x\to 0} \frac{e^x+e^{-x}-2}{\sin^{2}{x}} 15-2 \lim_{x\to 0} \frac{1+x\sin{x}-\cos{2x}}{\sin^{2}{x}}
15-3 \lim_{x\to -1} \frac{x^3+1}{\sin{(x+1)}} 15-4 \lim_{x\to a} \frac{\operatorname{tg}{x}-\operatorname{tg}{a}}{\ln{x}-\ln{a}}
15-5 \lim_{x\to 0} \frac{\sqrt{1+\operatorname{tg}{x}} -\sqrt{1+\sin {x}}} {x^3} 15-6 \lim_{x\to 0} \frac{e^{\alpha x}-e^{\beta x}}{\sin {\alpha x}-\sin {\beta x}}
15-7 \lim_{x\to 0} \frac{\sqrt{1+x\sin{x}} -1}{e^{x^2}-1} 15-8 \lim_{x\to 0} \frac{x^2\left(e^x-e^{-x}\right)}{e^{x^3+1}-e}
15-9 \lim_{x\to \frac{\pi}{3}} \frac{1-2\cos {x}}{\sin {(\pi -3x)}} 15-10 \lim_{x\to 1} \frac{1-x^2}{\sin {\pi x}}
15-11 \lim_{x\to \frac{\pi}{4}} \frac{\sin {x}-\cos {x}}{\ln {\operatorname{tg}{x}}} 15-12 \lim_{x\to b} \frac{a^x-a^b}{x-b}
15-13 \lim_{x\to 0} \frac{1-\cos {2x}+\operatorname{tg}^{2}{x}}{x\sin {3x}} 15-14 \lim_{x\to 0} \frac{\sin {2x}-2\sin {x}}{x\cdot \ln {\cos {5x}}}
15-15 \lim_{h\to 0} \frac{\ln {\left(x+h \right)} + \ln {\left(x-h \right)} - 2\ln {x}}{h^2},\; x>0 15-16 \lim_{x\to 1} \frac{1-x}{\log_{2} {x}}
15-17 \lim_{x\to 0} \frac{e^{\sin{2x}}-e^{\sin {x}}}{\operatorname{tg}{x}} 15-18 \lim_{x\to 1} \frac{2^x-2}{\ln {x}}
15-19 \lim_{h\to 0} \frac{\sin {(x+h)}-\sin {(x-h)}}{h} 15-20 \lim_{x\to 0} \frac{\sqrt {x+2} -\sqrt {2} }{\sin {3x}}
15-21 \lim_{h\to 0} \frac{a^{x+h}+a^{x-h}-2a^x}{h} 15-22 \lim_{x\to 0} \frac{1-\sqrt{\cos {x}}}{1-\cos {\sqrt {x}}}
15-23 \lim_{x\to 3} \frac{\sqrt[3]{5+x}-2}{\sin {\pi x}} 15-24 \lim_{x\to \frac{\pi}{6}} \frac{2\sin^{2}{x}+\sin {x}-1}{2\sin^{2}{x}-3\sin {x}+1}
15-25 \lim_{x\to 10} \frac{\lg {x}-1}{\sqrt {x-9} -1} 15-26 \lim_{x\to 0} \frac{3^{x+1}-3}{\ln {\left( 1+x\sqrt{1+x e^x}\right)}}
15-27 \lim_{x\to 0} \frac{\sqrt{\cos {x}}-1}{\sin^{2}{2x}} 15-28 \lim_{x\to 0} \frac{\sin {bx}-\sin {ax}}{\ln {\left(\operatorname{tg}{\left(\frac{\pi}{4} + ax\right)} \right)}}
15-29 \lim_{x\to \frac{\pi}{2}} \frac{1-\sin^{3}{x}}{\cos^{2}{x}} 15-30 \lim_{x\to 3} \frac{\log_{3} {x}-1}{\operatorname{tg}{\pi x}}
15-31 \lim_{x\to 1} \frac{e^x-e}{\sin {\left(x^2-1\right)}}

Задачи 16

Вычислить предел функции:

Задача Условие Задача Условие
16-1 \lim_{x\to 0} \left(1-\ln {\left(1+x^3 \right)} \right)^{\frac{3}{x^2\arcsin {x}}} 16-2 \lim_{x\to 0} \left(\cos {\sqrt {x}} \right)^{\frac{1}{x}}
16-3 \lim_{x\to 0} \left(\frac{1+x\cdot 2^x}{1+x\cdot 3^x}\right)^{\frac{1}{x^2}} 16-4 \lim_{x\to 0} \left(2-3^{\operatorname{arctg}^{2}{\sqrt{x}}} \right)^{\frac{2}{\sin {x}}}
16-5 \lim_{x\to 0} \left(\frac{1+\sin {x}\cdot \cos {\alpha x}} {1+\sin {x}\cdot \cos{\beta x}} \right)^{\operatorname{ctg}^{3}{x}} 16-6 \lim_{x\to 0} \left(5-\frac{4}{\cos {x}} \right)^{\frac{1}{\sin^{2}{3x}}}
16-7 \lim_{x\to 0} \left(1-\ln {\left(1+\sqrt[3]{x} \right)} \right)^{\frac{x}{\sin^{4}{\sqrt[3]{x}}}} 16-8 \lim_{x\to 0} \left(2-e^{\arcsin^{2}{\sqrt{x}}} \right)^{\frac{3}{x}}
16-9 \lim_{x\to 0} \left(\cos {\pi x} \right)^{\frac{1}{x\sin {\pi x}}} 16-10 \lim_{x\to 0} \left(1+\sin^{2}{3x}\right)^{\frac{1}{\ln {\cos {x}}}}
16-11 \lim_{x\to 0} \left(\operatorname{tg}{\left(\frac{\pi}{4}-x \right)} \right)^{\operatorname{ctg}{x}} 16-12 \lim_{x\to 0} \left(1-x\sin^{2}{x} \right)^{\frac{1}{\ln {\left(1+\pi x^3\right)}}}
16-13 \lim_{x\to 0} \left(2-5^{\arcsin {x^3}} \right)^{\frac{\operatorname{cosec}^{2}{x}} {x}} 16-14 \lim_{x\to 0} \left(2-\cos {3x} \right)^{\frac{1}{\ln {\left(1+x^2 \right)}}}
16-15 \lim_{x\to 0} \left(2-e^{\sin{x}} \right)^{\operatorname{ctg}{\pi x}} 16-16 \lim_{x\to 0} \left(\cos {x} \right)^{\frac{1} {\ln {\left(1+\sin^{2}{x} \right)}}}
16-17 \lim_{x\to 0} \left(2-e^{x^2} \right)^{1 / \ln {\left(1+\operatorname{tg}^{2} {\left(\frac{\pi x}{3} \right)} \right)}} 16-18 \lim_{x\to 0} \left(3-2\cos{x} \right)^{-\operatorname{cosec}^{2}{x}}
16-19 \lim_{x\to 0} \left(2-3^{\sin^{2}{x}} \right)^{\frac{1}{\ln {(\cos {x})}}} 16-20 \lim_{x\to 0} \sqrt[x^2]{2-\cos {x}}
16-21 \lim_{x\to 0} \left(6-\frac{5}{\cos {x}}\right)^{\operatorname{ctg}^{2}{x}} 16-22 \lim_{x\to 0} \left(3-\frac{2}{\cos {x}}\right)^{\operatorname{cosec}^{2}{x}}
16-23 \lim_{x\to 0} \left(\frac{1+\sin{x} \cdot \cos {2x}}{1+\sin {x}\cdot \cos {3x}}\right)^{\frac{1}{\sin {x^3}}} 16-24 \lim_{x\to 0} \left(2-e^{x^2} \right)^{\frac{1}{1-\cos {\pi x}}}
16-25 \lim_{x\to 0} \left(1+\ln {\frac{1}{3}}\cdot \operatorname{arctg}^{6}{\sqrt{x}}\right)^{\frac{1}{x^3}} 16-26 \lim_{x\to 0} \left(\frac{1+\operatorname{tg}{x}\cdot \cos {2x}}{1+\operatorname{tg}{x}\cdot \cos {5x}} \right)^{\frac{1}{x^3}}
16-27 \lim_{x\to 0} \left(\frac{1+x\cdot 3^x}{1+x\cdot 7^x} \right)^{\frac{1}{\operatorname{tg}^{2}{x}}} 16-28 \lim_{x\to 0} \left(1+\operatorname{tg}^{2}{x} \right)^{\frac{1}{\ln \left(1+3x^2 \right)}}
16-29 \lim_{x\to 0} \left(1-\ln {(\cos{x})} \right)^{\frac{1}{\operatorname{tg}^{2}{x}}} 16-30 \lim_{x\to 0} \left(1-\sin^{2}{\frac{x}{2}}\right)^{\frac{1}{\ln \left(1+\operatorname{tg}^{2}{3x} \right)}}
16-31 \lim_{x\to 0} \left(\frac{1+x^2\cdot 2^x}{1+x^2\cdot 5^x} \right)^{\frac{1}{\sin^{3}{x}}}
Источник — «http://pluspi.org/wiki/index.php/Zadachnik_Kuznecova_Predely_Zadachi_13-16»
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