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Question
\[\lim_{x \to \infty} \left[ \frac{x^4 + 7 x^3 + 46x + a}{x^4 + 6} \right]\] where a is a non-zero real number.
Solution
\[\lim_{x \to \infty} \left[ \frac{x^4 + 7 x^3 + 46x + a}{x^4 + 6} \right]\] Dividing the numerator and the denominator by x4:
\[\lim_{x \to \infty} \left[ \frac{1 + \frac{7}{x} + \frac{46}{x^2} + \frac{9}{x^4}}{1 + \frac{6}{x^4}} \right]\]
\[\text{ As } x \to \infty , \frac{1}{x}, \frac{1}{x^2}, \frac{1}{x^3}, \frac{1}{x^4} \to 0\]
\[ = \frac{1}{1}\]
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