Question

lim/X,=infinity. ✓3x^2-2 + ✓2x^2+5/5x-3​

Answer 1

$\large\underline\purple{\bold{Solution :- }}$

$\bf \: \displaystyle \rm \ \lim_{ x\to \infty} \dfrac{ \sqrt{ {3x}^{2} - 2 } + \sqrt{ {2x}^{2} + 5 } }{5x - 3}$

$\bf \: = \displaystyle \rm \ \lim_{ x\to \infty} \dfrac{ \sqrt{ {x}^{2}\bigg(3 - \dfrac{2}{ {x}^{2} } \bigg)} + \sqrt{ {x}^{2}\bigg(2 + \dfrac{5}{ {x}^{2} } \bigg) } }{x\bigg(5 - \dfrac{3}{x} \bigg)}$

$\bf \: = \displaystyle \rm \ \lim_{ x\to \infty} \dfrac{x \sqrt{ \bigg(3 - \dfrac{2}{ {x}^{2} } \bigg)} + x\sqrt{ \bigg(2 + \dfrac{5}{ {x}^{2} } \bigg) } }{x\bigg(5 - \dfrac{3}{x} \bigg)}$

$\bf \: = \displaystyle \rm \ \lim_{ x\to \infty} \dfrac{ x \bigg(\sqrt{ \bigg(3 - \dfrac{2}{ {x}^{2} } \bigg)} + \sqrt{ \bigg(2 + \dfrac{5}{ {x}^{2} } \bigg) } \bigg)}{x\bigg(5 - \dfrac{3}{x} \bigg)}$

$\bf \: = \displaystyle \rm \ \lim_{ x\to \infty} \dfrac{ \sqrt{ \bigg(3 - \dfrac{2}{ {x}^{2} } \bigg)} + \sqrt{ \bigg(2 + \dfrac{5}{ {x}^{2} } \bigg) } }{\bigg(5 - \dfrac{3}{x} \bigg)}$

$\bf \:  = \: \dfrac{ \sqrt{3 - 0} + \sqrt{2 - 0} }{5 - 0}$

$\bf \:  = \dfrac{ \sqrt{3} + \sqrt{2} }{5}$