First slide

Continuity

Question

$f (x) \{\begin{cases} \begin{matrix} \frac{\cos 3 x - \cos 4 x}{x^{2}}, & f o r x \neq 0 \end{matrix} \\ \begin{matrix} \frac{7}{2} & , f o r x = 0 \end{matrix} \end{cases} a t x = 0 i s$

Moderate

A

continuous
B

discontinuous
C

left continuous
D

right continuous

Solution

$\lim_{x \to 0} f (x)$
$= \lim_{x \to 0} \frac{\cos a x - \cos b x}{x^{2}} = \frac{b^{2} - a^{2}}{2} = \lim_{x \to 0} \frac{\cos 3 x - \cos 4 x}{x^{2}} = \frac{4^{2} - 3^{2}}{2} = \frac{7}{2} = f (0)$

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