a)

$\underset{x\to -\infty }{\lim }{x}^{25}=-\infty$  

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b)

$\underset{x\to -\infty }{\lim }\frac{{\left(-x^3\right)}^2}{x^5}=\underset{x\to -\infty }{\lim }\frac{x^6}{x^5}\stackrel{\left(\ast \right)}{=}\underset{x\to -\infty }{\lim }x=-\infty$  

${}^{\left(\ast \right)\text{Skoro}x\to -\infty ,\text{to moz˙emy przyjącˊ, z˙e}x\ne 0}$  

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c)

$\underset{x\to +\infty }{\lim }\frac{\sqrt{x}}{x}=\underset{x\to +\infty }{\lim }\frac{1}{\underset{\underset{+\infty }{\downarrow }}{\underbrace{\sqrt{x}}}}=0$  

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d) 

$\underset{x\to +\infty }{\lim }\frac{\sqrt{x}}{\sqrt[3]{x}}=\underset{x\to +\infty }{\lim }\frac{x^{\frac{1}{2}}}{x^{\frac{1}{3}}}=\underset{x\to +\infty }{\lim }\frac{x^{\frac{3}{6}}}{x^{\frac{2}{6}}}=\underset{x\to +\infty }{\lim }{x}^{\frac{1}{6}}=\underset{x\to +\infty }{\lim }\sqrt[6]{x}=+\infty$