LimitsEasy

Indeterminate form ∞/∞ — rational

Calculate lim_{x→+∞} (3x² − 2x + 1)/(x² + 5).

Given data

lim_{x→+∞} (3x² − 2x + 1)/(x² + 5)

Steps

0 / 3

Factor out the dominant term x² in the numerator and denominator.
Write your working — the numerical result must appear in the text (logica)
As x→+∞, what do 1/x, 1/x² and 5/x² equal?
What is the value of the limit after simplification?

Full worked solution

Factor out the dominant term x² in the numerator and denominator.
$\dfrac{x^2(3 - 2/x + 1/x^2)}{x^2(1 + 5/x^2)}$
$\frac{3x^2 - 2x + 1}{x^2 + 5} = \frac{x^2(3 - 2/x + 1/x^2)}{x^2(1 + 5/x^2)} = \frac{3 - 2/x + 1/x^2}{1 + 5/x^2}$ . Factor out $x^2$ , the dominant term.
As x→+∞, what do 1/x, 1/x² and 5/x² equal?
$\dfrac{2}{x} \to 0,\quad \dfrac{1}{x^2} \to 0,\quad \dfrac{5}{x^2} \to 0$
As $x \to +\infty$ , $2/x \to 0$ , $1/x^2 \to 0$ , and $5/x^2 \to 0$ . All terms with $x$ in the denominator vanish at infinity.
What is the value of the limit after simplification?
$\dfrac{3 - 0 + 0}{1 + 0} = 3$
$\displaystyle\lim_{x\to +\infty} \frac{3x^2 - 2x + 1}{x^2 + 5} = \frac{3 - 0 + 0}{1 + 0} = \mathbf{3}$ . The limit equals the ratio of the leading coefficients when numerator and denominator have the same degree.

Result:lim_{x→+∞} (3x² − 2x + 1)/(x² + 5) = 3.

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