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límite cuando x tiende a infinity de x(x-sqrt(x^2-4))

Solução

x \to \infty lim (x (x - x^{2} - 4))

2

Passos da solução

x \to \infty lim (x (x - x^{2} - 4))

Multiplicar pelo conjugado de

x - x^{2} - 4 : \frac{4}{x + x ^{2} - 4}

= x \to \infty lim (x \frac{4}{x + x ^{2} - 4})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 4 \cdot x \to \infty lim (x \frac{1}{x + x ^{2} - 4})

Dividir pelo denominador de maior potência:

\frac{\frac{1}{x}}{1 + \frac{x ^{2} - 4}{x}}

= 4 \cdot x \to \infty lim (x \frac{\frac{1}{x}}{1 + \frac{x ^{2} - 4}{x}})

Simplificar

x \frac{\frac{1}{x}}{1 + \frac{x ^{2} - 4}{x}} : \frac{x}{x + x ^{2} - 4}

= 4 \cdot x \to \infty lim (\frac{x}{x + x ^{2} - 4})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{1}{1 + \frac{x}{x ^{2} - 4}})

Simplificar

\frac{1}{1 + \frac{x}{x ^{2} - 4}} : \frac{x ^{2} - 4}{x ^{2} - 4 + x}

= 4 \cdot x \to \infty lim (\frac{x ^{2} - 4}{x ^{2} - 4 + x})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1})

Simplificar

\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1} : \frac{x}{x + x ^{2} - 4}

= 4 \cdot x \to \infty lim (\frac{x}{x + x ^{2} - 4})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{1}{1 + \frac{x}{x ^{2} - 4}})

Simplificar

\frac{1}{1 + \frac{x}{x ^{2} - 4}} : \frac{x ^{2} - 4}{x ^{2} - 4 + x}

= 4 \cdot x \to \infty lim (\frac{x ^{2} - 4}{x ^{2} - 4 + x})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1})

Simplificar

\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1} : \frac{x}{x + x ^{2} - 4}

= 4 \cdot x \to \infty lim (\frac{x}{x + x ^{2} - 4})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{1}{1 + \frac{x}{x ^{2} - 4}})

Simplificar

\frac{1}{1 + \frac{x}{x ^{2} - 4}} : \frac{x ^{2} - 4}{x ^{2} - 4 + x}

= 4 \cdot x \to \infty lim (\frac{x ^{2} - 4}{x ^{2} - 4 + x})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1})

Simplificar

\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1} : \frac{x}{x + x ^{2} - 4}

= 4 \cdot x \to \infty lim (\frac{x}{x + x ^{2} - 4})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{1}{1 + \frac{x}{x ^{2} - 4}})

Simplificar

\frac{1}{1 + \frac{x}{x ^{2} - 4}} : \frac{x ^{2} - 4}{x ^{2} - 4 + x}

= 4 \cdot x \to \infty lim (\frac{x ^{2} - 4}{x ^{2} - 4 + x})

Aplicar a regra de L'Hopital

= 4 \cdot x \to \infty lim (\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1})

Simplificar

\frac{\frac{x}{x ^{2} - 4}}{\frac{x}{x ^{2} - 4} + 1} : \frac{x}{x + x ^{2} - 4}

= 4 \cdot x \to \infty lim (\frac{x}{x + x ^{2} - 4})

Dividir pelo denominador de maior potência:

\frac{1}{1 + \frac{x ^{2} - 4}{x}}

= 4 \cdot x \to \infty lim (\frac{1}{1 + \frac{x ^{2} - 4}{x}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

Com exceção da forma indeterminada

= 4 \cdot \frac{lim _{x \to \infty} ( 1 )}{lim _{x \to \infty} ( 1 + \frac{x ^{2} - 4}{x} )}

x \to \infty lim (1) = 1

x \to \infty lim (1 + \frac{x ^{2} - 4}{x}) = 2

= 4 \cdot \frac{1}{2}

Simplificar

4 \cdot \frac{1}{2} : 2

= 2

t an g e n t 2 \frac{1}{x}, at x = 3

\int \frac{9 x ^{2} - 1}{x} d x

\int (1 - 2 e^{t a n (x)}) sec^{2} (x) d x

x \to 4 - lim (\frac{x + 2}{x - 4})

x \to 0 lim (\frac{lo g _{e} ( 1 + 3 x ^{2} )}{x ( e ^{x} - 1 )})