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

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

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

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

Ejercicio 4
$$\ f(x)= x*e^{-x}$$ $$\ f'(x)= e^{-x}(1-x)$$

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