Problems from Indeterminate form 0 x infinity

Calculate the following limit:

$$\displaystyle\lim_{x \to{+}\infty}{\dfrac{x+1}{3x^2}\cdot\dfrac{x^2}{x-1}}$$

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$$\displaystyle\lim_{x \to{+}\infty}{\dfrac{x+1}{3x^2}\cdot\dfrac{x^2}{x-1}}=0\cdot\infty$$

As this limit $$x+1\approx x$$ y $$x-1\approx x$$

$$\displaystyle\lim_{x \to{+}\infty}{\dfrac{x\cdot x^2}{3x^2\cdot x}}=\dfrac{1}{3}$$

$$\dfrac{1}{3}$$

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Calculate the following limit:

$$\displaystyle\lim_{x \to{+}\infty}{\dfrac{\sqrt{x}}{x^2}\cdot(x^3+1)}$$

See development and solution

$$\displaystyle\lim_{x \to{+}\infty}{\dfrac{\sqrt{x}}{x^2}\cdot(x^3+1)}=0\cdot\infty$$

As this limit $$x^3+1\approx x^3$$

$$\displaystyle\lim_{x \to{+}\infty}{\dfrac{x^3\sqrt{x}}{x^2}}=\lim_{x \to{+}\infty}{x\sqrt{x}}=+\infty$$

$$+\infty$$

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