Evaluate the following limits: _ x 0 ( x-1 x^2 )

Question

Evaluate the following limits:
$\lim _{x \rightarrow 0}\left(\frac{\sec x-1}{x^2}\right)$

✓

Answer

$ \lim _{x \rightarrow 0} \frac{\sec x-1}{x^2}$
$=\lim _{x \rightarrow 0} \frac{(\sec x-1)(\sec x+1)}{x^2(\sec x+1)}$
$=\lim _{x \rightarrow 0} \frac{\sec ^2 x-1}{x^2(\sec x+1)}=\lim _{x \rightarrow 0} \frac{\tan ^2 x}{x^2(\sec x+1)}$
$=\lim _{x \rightarrow 0}\left[\left(\frac{\tan x}{x}\right)^2 \times \frac{1}{\sec x+1}\right]$
$=\lim _{x \rightarrow 0}\left(\frac{\tan x}{x}\right)^2 \times \lim _{x \rightarrow 0} \frac{1}{\sec x+1}$
$=(1)^2 \times \frac{1}{1+1} \quad \ldots\left[\because \lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta}=1\right]$
$=\frac{1}{2} $

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