Question

Evaluate: ${lim}_{x\to \infty }(\sqrt{25x^2-3x}+5x)$

Answer 1

$\begin{array}{c}{lim}_{x\to \infty }\left(\sqrt{25x^2-3x}+5x\right)\\ Byrationalisation\\ \Rightarrow {lim}_{x\to \infty }\frac{\left(\sqrt{25x^2-3x}+5x\right)\left(\sqrt{25x^2-3x}+5x\right)}{\left(\sqrt{25x^2-3x}-5x\right)}\\ \Rightarrow {lim}_{x\to \infty }\frac{25x^2-3x-25x^2}{x\left[\left(25-\frac{3}{x}\right)-5\right]}\\ \Rightarrow {lim}_{x\to \infty }\frac{-3x}{x\sqrt{\left(25-\frac{3}{x}\right)-5}}\\ \Rightarrow {lim}_{x\to \infty }\frac{-3}{\sqrt{25-0}-5}=\frac{-3}{\pm 5-5}=\frac{-3}{-10}=\frac{3}{10}\end{array}$