A= lim_"x->oo" sqrt((x^2+4)/(x+4)) B=lim_"x->oo" sqrt((x^2+5)/x^3) Find A and B ?

Sep 28, 2016

See below

Explanation:

lim_ (x -> oo) sqrt((x^2+4)/(x+4)

= lim_ (x -> oo) sqrt((1+4/x^2)/(1/x+4/x^2)

= sqrt((1+lim_ (x -> oo)4/x^2)/(lim_ (x -> oo)1/x+ lim_ (x -> oo)4/x^2)

$= \sqrt{\frac{1 + 0}{0 + 0}}$

$= \infty$

${\lim}_{x \to \infty} \sqrt{\frac{{x}^{2} + 5}{x} ^ 3}$

$= {\lim}_{x \to \infty} \sqrt{\frac{1}{x} + \frac{5}{x} ^ 3}$

$= \sqrt{{\lim}_{x \to \infty} \frac{1}{x} + {\lim}_{x \to \infty} \frac{5}{x} ^ 3}$

$= \sqrt{0}$

$= 0$

Sep 28, 2016

$\infty$ and $0$

Explanation:

${\lim}_{x \to \infty} \sqrt{\frac{{x}^{2} + 4}{x + 4}} = \sqrt{{\lim}_{x \to \infty} \left({x}^{2} / x\right) \frac{1 + \frac{4}{x} ^ 2}{1 + \frac{4}{x}}} = \infty$

${\lim}_{x \to \infty} \sqrt{\frac{{x}^{2} + 5}{x} ^ 3} = \sqrt{{\lim}_{x \to \infty} \frac{{x}^{2}}{{x}^{3}} \left(1 + \frac{5}{x} ^ 2\right)} = 0$

Sep 28, 2016

$\infty , 0$

Explanation:

$A = {\lim}_{\text{x->oo}} \sqrt{\frac{{x}^{2} + 4}{x + 4}}$

$= {\lim}_{\text{x->oo}} \sqrt{\frac{x + \frac{4}{x}}{1 + \frac{4}{x}}}$

$= {\lim}_{\text{x->oo}} \sqrt{\frac{x}{1}} = \infty$

$B = {\lim}_{\text{x->oo}} \sqrt{\frac{{x}^{2} + 5}{x} ^ 3}$

$= {\lim}_{\text{x->oo}} \sqrt{\frac{1 + \frac{5}{x} ^ 2}{x}}$

$= {\lim}_{\text{x->oo}} \sqrt{\frac{1}{x}} = 0$