Problems from Indeterminate form infinity minus infinity

Find the following limits:

a) $lim_{x \to + \infty} (\frac{3 x + 5}{2} - \frac{x^{2} - 2}{x})$

b) $lim_{x \to + \infty} (\sqrt{x + 1} - \sqrt{x + 2})$

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Development:

a) $lim_{x \to + \infty} (\frac{3 x + 5}{2} - \frac{x^{2} - 2}{x}) = lim_{x \to + \infty} (\frac{x (3 x + 5)}{2 x} - \frac{2 (x^{2} - 2)}{2 x}) =$ $= lim_{x \to + \infty} (\frac{3 x^{2} + 5 x - 2 x^{2} + 4}{2 x}) = lim_{x \to + \infty} \frac{x}{2} = + \infty$

b) $lim_{x \to + \infty} (\sqrt{x + 1} - \sqrt{x + 2}) = lim_{x \to + \infty} \frac{(x + 1) - (x + 2)}{\sqrt{x + 1} + \sqrt{x + 2}} =$ $= lim_{x \to + \infty} \frac{- 1}{\sqrt{x + 1} + \sqrt{x + 2}} = 0$ where we have applied the formula for the squared roots.

Solution:

a) $+ \infty$

b) $0$

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