Question

If the function defined by $f\left(x\right)=\frac{\tan 2\pi x+\sin \frac{\pi x}{2}+\tan \frac{\pi x}{2}}{x^2+4x-12}$ is continuous at $x=2$ then $f^{\prime }\left(2\right)$

• A. equals $\frac{\pi }{4}$
• B. equals $\frac{3\pi }{8}$
• C. equals $2\pi$
• D. is non existent

Answer 1

The correct option is A equals $\frac{\pi }{4}$

Substituting gives indeterminate form.
Applying L Hospital's Rule,
$=\frac{2\pi {\sec }^{2}2\pi x+\frac{\pi }{2}\cos \frac{\pi x}{2}+\frac{\pi }{2}{\sec }^2\frac{\pi x}{2}}{2x+4}$
Substituting,
$=\frac{2\pi -\frac{\pi }{2}+\frac{\pi }{2}}{2x+4}$

$=\frac{\pi }{4}$