First slide

Introduction to limits

Question

The value of $lim_{x \to \infty} \frac{a x^{2} + b x + c}{d x^{2} + c x + f}$ , is

Moderate

A

$\frac{a}{d}$
B

$\frac{d}{a}$
C

$\frac{b}{e}$
D

$\frac{c}{f}$

Solution

Here, the expression assumes the form $\frac{\infty}{\infty}$ .
We notice that the highest power of $x$ in both the numerator and denominator
is 2. So, we divide each term in both the numerator and denominator by $x^{2}$ .
$∴ lim_{x \to \infty} \frac{a x^{2} + b x + c}{d x^{2} + e x + f} = lim_{x \to \infty} \frac{a + \frac{b}{x} + \frac{c}{x^{2}}}{d + \frac{e}{x} + \frac{f}{x^{2}}} = \frac{a + 0 + 0}{d + 0 + 0} = \frac{a}{d}$

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