First slide

Introduction to limits

Question

$If f (x) = |\begin{array}{ccc} \sin x & \cos x & \tan x \\ x^{3} & x^{2} & x \\ 2 x & 1 & x \end{array}| then lim_{x \to 0} \frac{f (x)}{x^{2}} =$

Moderate

Solution

$f (x) = |\begin{array}{ccc} \sin x & \cos x & \tan x \\ x^{1} & x^{2} & x \\ 2 x & 1 & x \end{array}|$
$d i v i d e L H S b y x^{2} a n d C_{1} a n d C_{3} b y x i n R H S$
$∴ \frac{f (x)}{x^{2}} = |\begin{array}{ccc} \frac{\sin x}{x} & \cos x & \frac{\tan x}{x} \\ x^{2} & x^{2} & 1 \\ 2 & 1 & 1 \end{array}|$
$∴ lim_{x \to 0} \frac{f (x)}{x^{2}} = |\begin{array}{lll} 1 1 1 \\ 0 0 1 \\ 2 1 1 \end{array}| = 1$

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