Question

LIMITS CHAPTER PROBLEMATI

Answer 1

$\lim_{x \to \infty} \bigl ( x - \sqrt{x^2 + x} \bigr ) \\ \\ Multiply\ and\ divide\ by\ conjugate \\ \\ \to \lim_{x \to \infty} \left(\frac{x}{x+\sqrt{x^2+x}}\right) \\ \\ Now, Dividing \ by\ highest\ power\ of\ x. \\ \\ \to -\lim_{x \to \infty} \left(\frac{1}{1+\sqrt{1+\frac{1}{x}}}\right) \\ \\ \to -\frac{\lim _{x\to \infty \:}\left(1\right)}{\lim _{x\to \infty \:}\left(1+\sqrt{1+\frac{1}{x}}\right)} \\ \\ \to \frac{-1}{2}$