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Question
\[\lim_{x \to \infty} \frac{3 x^{- 1} + 4 x^{- 2}}{5 x^{- 1} + 6 x^{- 2}}\]
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Solution
\[\lim_{x \to \infty} \left[ \frac{3 x^{- 1} + 4 x^{- 2}}{5 x^{- 1} + 6 x^{- 2}} \right]\]
\[ = \lim_{x \to \infty} \left[ \frac{\frac{3}{x} + \frac{4}{x^2}}{\frac{5}{x} + \frac{6}{x^2}} \right]\]
\[ = \lim_{x \to \infty} \left[ \frac{3x + 4}{5x + 6} \right]\]
\[\text{ Dividingthe numerator and the denominator by } x:\]
\[ \lim_{x \to \infty} \left[ \frac{3 + \frac{4}{x}}{5 + \frac{6}{x}} \right] = \frac{3}{5}\]
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