First slide

Introduction to limits

Question

Let $α$ and $β$ be the roots of $a x^{2} + b x + c = 0,$ then $\underset{x \to α}{L t} \frac{1 - \cos (a x^{2} + b x + c)}{{(x - α)}^{2}} =$

Moderate

A

$\frac{a^{2} {(α - β)}^{2}}{2}$
B

$\frac{a^{2}}{2 {(α - β)}^{2}}$
C

$\frac{a^{2}}{{(α - β)}^{2}}$
D

$- \frac{a^{2}}{2 {(α - β)}^{2}}$

Solution

$a x^{2} + b x + c = a (x - α) (x - β)$ given limit is $\underset{x \to α}{L i m} \frac{2 \sin^{2} {(\frac{a (x - α) (x - β)}{2})}^{2}}{{(x - α)}^{2}} = \frac{a^{2} {(α - β)}^{2}}{2}$

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