User:"Jose Freyle"

LIMITES

En esta sección encontraremos la solución de algunos problemas relacionados con limites al infinito, trigonométricos y algunos que se demuestran directamente.

Ejercicio 1

$\lim _{x\to 0}{\frac {sen(x)}{3x}}$
 =${\frac {1}{3}}\lim _{x\to 0}{\frac {3}{1}}{\frac {sen(x)}{x}}$
 =${\frac {1}{3}}*1$
 =${\frac {1}{3}}$

Ejercicio 2

$\lim _{x\to 0}{\frac {sen(x)}{x}}$
 =$\lim _{x\to 0}{\frac {5}{1}}{\frac {sen(5x)}{5x}}$
 =${\frac {5*1}{1}}$
 =$\ 5(1)$
 =$\ 5$

Ejercicio 3

$\lim _{x\to 0}{\frac {(x^{3}+16x)}{x^{2}+4x}}$
 =$\lim _{x\to 0}{\frac {x(x^{2}+16)}{x(x+4)}}$
 =$\lim _{x\to 0}{\frac {x^{2}+16}{x+4}}$
 =${\frac {0^{2}+16}{0+4}}$
 =${\frac {16}{4}}$
 =$\ 4$

Ejercicio 4

$\lim _{x\to 1}{\frac {x^{2}+3x-4}{x-1}}$
 =$\lim _{x\to 1}{\frac {(x-4)(x-1)}{x-1}}$
 =$\ 1+4$
 =$\ 5$

Ejercicio 5

$\lim _{x\to 3}{\frac {2}{x}}+1$
 =$\lim _{x\to 3}{\frac {2+x}{x}}$
 =${\frac {2+3}{3}}$
 =${\frac {5}{3}}$

Ejercicio 6

$\lim _{x\to 1}{\frac {5x-x^{2}}{x^{2}+2x-4}}$
 =${\frac {5(1)-(1)^{2}}{12+2(1)-4}}$
 =${\frac {5-1}{1+2-4}}$
 =${\frac {6}{-1}}$
 =$\ -4$

Ejercicio 7

$\lim _{x\to 1}{\frac {x^{2}-x}{2x^{+}5x-7}}$
 =$\lim _{x\to 1}{\frac {x(x-1)}{7(x-1)}}$
 =${\frac {1}{7}}$

Ejercicio 8

$\lim _{x\to -1}{\frac {x^{2}-1}{x^{2}+3x+2}}$
 =$\lim _{x\to -1}{\frac {(x+1)(x-1)}{(x+2)(x+1)}}$
 =${\frac {-1-1}{-1+2}}$
 =${\frac {-2}{1}}$
 =$\ -2$

Ejercicio 9

$\lim _{x\to 2}{\frac {3x^{5}-8x^{4}+5x^{3}-3x-2}{7x^{2}-6x-16}}$
=$\lim _{x\to 2}{\frac {(x-2)(3x^{4}-2x^{3}+x^{2}+2x+1)}{(x-2)(7x+8)}}$
=${\frac {48-16+4+4+1}{14+8}}$
=${\frac {41}{22}}$

Ejercicio 10

 $\lim _{x\to \infty }{\frac {x^{3}-2x+1}{x^{2}+1}}$
=$\lim _{x\to \infty }{\frac {{\frac {x^{3}}{x^{2}}}-{\frac {2x}{x^{2}}}+{\frac {1}{x^{2}}}}{{\frac {x^{2}}{x^{2}}}+{\frac {1}{x^{2}}}}}$
=${\frac {x-0+0}{1+0}}$
=${\frac {x}{1}}$
=$\infty$

Ejercicio 11

$\lim _{x\to \infty }{\frac {5}{x^{2}+8x+15}}$
=$\lim _{x\to \infty }{\frac {\frac {5}{x^{2}}}{{\frac {x^{2}}{x^{2}}}+{\frac {8x}{x^{2}}}+{\frac {15}{x^{2}}}}}$
=${\frac {0}{1+0+0}}$
=${\frac {0}{1}}$
=$\ 0$

Ejercicio 12

$\lim _{x\to \infty }{\frac {1}{x^{2}+5x-6}}$
=$\lim _{x\to \infty }{\frac {\frac {1}{x^{2}}}{{\frac {x^{2}}{x^{2}}}+{\frac {5x}{x^{2}}}-{\frac {6}{x^{2}}}}}$
=${\frac {0}{1+0-0}}$
=${\frac {0}{1}}$
=$\ 0$

Ejercicio 13

$\lim _{t\ 0}{\frac {(}{\sqrt[{2}]{2t}}})-{\sqrt[{2}]{2}}){t}$
$\lim _{t\ 0}{\frac {{\sqrt[{2}]{2t}}-{\sqrt[{2}]{2}}}{t}}*{\sqrt[{2}]{2t}}+{\sqrt[{2}]{2}}{\sqrt[{2}]{2t}}+{\sqrt[{2}]{\sqrt[{2}]{2}}}$
$\lim _{t\ 0}{\frac {-t}{t({\sqrt[{2}]{2t}}+{\sqrt[{2}]{)}}}}$
$\lim _{t\ 0}{\frac {-1}{({\sqrt[{2}]{2t}}+{\sqrt[{2}]{)}}}}$
${\frac {-1}{({\sqrt[{2}]{2*0}}+{\sqrt[{2}]{)}}}}$
${\frac {-1}{({0}+{\sqrt[{2}]{2}})}}$
${\frac {-1}{({\sqrt[{2}]{2}})}}$

Ejercicio 14

$\lim _{x\to \infty }\ ((x^{5}+7x^{4}+2)^{(}c)-x)$
$\lim _{x\to \infty }{\frac {(x^{5}+7x^{4}+2)^{(}2c)}{(x^{5}+7x^{4}+2)^{(}c)+x}}$

$\lim _{x\to \infty }\ ((x^{5}+7x^{4}+2)^{(}{\frac {1}{5}})-x)$

${\frac {7}{5}}$

DERIVADAS

Ejercicio 1

$\ y=(3x^{5}-1)(2x+3)$
$\ y'=(3x^{5}-1)(2)+(2x+3)(15x^{4})$
$\ y'=6x^{5}-2+30x^{5}+45x^{4}$
$\ y'=36x^{5}+45x^{4}-2$

Ejercicio 2

$\ y=6x^{4}+3x^{3}+6x^{2}-8x+2$
$\ y'=24x^{3}+9x^{2}+12x-8$

Ejercicio 3

$\ f(x)=cos(x)$
$f'(x)=lim_{h\to 0}{\frac {f(x+h)-f(x)}{h}}$
$f'(x)=lim_{h\to 0}{\frac {cos(x+h)-cosx}{h}}$
$\lim _{h\to 0}{\frac {cos(x)*cos(h)-sen(x)*sen(h)-cos(x)}{h}}$
$\lim _{h\to 0}{\frac {cos(x)(cosh-1)}{h}}*{\frac {sen(x)sen(h)}{h}}$
$\cos(x)*lim_{h\to 0}{\frac {cos(h)-1}{h}}-sen(lim_{h\to 0}{\frac {sen(h)}{h}}$
$\lim _{h\to 0}\ 0*cos(x)-lim_{h\to 0}\ 1*-sen(x)$
$\ 0-sen(x)$
$\ -sen(x)$

Ejercicio 4

$\ f(x)=x*e^{-x}$
$\ f'(x)=e^{-x}(1-x)$

Ejercicio 5

$\ f(x)=5x^{5}+4x^{4}+3x^{3}+2x^{2}+x$
$\ f'(x)=25x^{4}+16x^{3}+9x^{2}+4x+1$