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A ratio far out

CalculusEasy

The ratio $\frac{4 x ^{2} - 3 x + 1}{2 x ^{2} + 5}$ settles to a fixed number as $x$ grows without bound. What is that number? Give an integer.

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Hint

Divide the numerator and the denominator by the highest power of $x$ .

Solution

Dividing the numerator and the denominator by $x^{2}$ gives $\frac{4 - \frac{3}{x} + \frac{1}{x ^{2}}}{2 + \frac{5}{x ^{2}}}$ .
Each term with $x$ in its denominator tends to $0$ .
So the limit is $\frac{4}{2} = 2$ .

Answer: $2$