Find the following limits:
a) $$\displaystyle\lim_{x \to{+}\infty}{\Big(\dfrac{3x+5}{2}-\dfrac{x^2-2}{x}\Big)}$$
b) $$\displaystyle\lim_{x \to{+}\infty}{(\sqrt{x+1}-\sqrt{x+2})}$$
See development and solution
Development:
a) $$$\lim_{x\to{+}\infty}{\Big(\dfrac{3x+5}{2}-\dfrac{x^2-2}{x}\Big)}=\lim_{x\to{+}\infty}{\Big(\dfrac{x(3x+5)}{2x}-\dfrac{2(x^2-2)}{2x}\Big)}=$$$ $$$=\lim_{x\to{+}\infty}{\Big(\dfrac{3x^2+5x-2x^2+4}{2x}\Big)}=\lim_{x\to{+}\infty}{\dfrac{x}{2}}=+\infty$$$
b) $$$\lim_{x\to{+}\infty}{(\sqrt{x+1}-\sqrt{x+2})}=\lim_{x\to{+}\infty}{\dfrac{(x+1)-(x+2)}{\sqrt{x+1}+\sqrt{x+2}}}=$$$ $$$=\lim_{x\to{+}\infty}{\dfrac{-1}{\sqrt{x+1}+\sqrt{x+2}}}=0$$$ where we have applied the formula for the squared roots.
Solution:
a) $$+\infty$$
b) $$0$$