Question

The value of $\underset{\theta \to 90^{\circ }}{\lim }\tan \theta -\sec \theta$ is

• A. 0
• B. -1
• C. 1
• D. None of these

Answer 1

Answer: (A)

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