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  4. The value of displaystyle lim θ arrow 90° tan θ- sec θ is

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Q. The value of $\displaystyle\lim _{\theta \rightarrow 90^{\circ}} \tan \theta-\sec \theta$ is

Limits

Solution:

$\displaystyle\lim _{\theta \rightarrow 90^{\circ}} \tan \theta-\sec \theta =\displaystyle\lim _{\theta \rightarrow 90^{\circ}} \frac{\sin \theta}{\cos \theta}-\frac{1}{\cos \theta} $
$ =\displaystyle\lim _{\theta \rightarrow 90^{\circ}} \frac{\sin \theta-1}{\cos \theta} \times \frac{\sin \theta+1}{\sin \theta+1} $
$ =\displaystyle\lim _{\theta \rightarrow 90^{\circ}} \frac{\sin ^2 \theta-1}{\cos \theta(\sin \theta+1)} $
$ =\displaystyle\lim _{\theta \rightarrow 90^{\circ}} \frac{-\cos ^2 \theta}{\cos \theta(\sin \theta+1)}$
$=\displaystyle\lim _{\theta \rightarrow 90^{\circ}} \frac{-\cos \theta}{\sin \theta+1}=0$