Thread: continuous at infinty?

1. continuous at infinty?

show that f is continuous at (-infinity, infinity)

f(x)= { sin(x) if x < (pi/4)
cos(x) if x >or equal to (pi/4)

confused???

2. Originally Posted by thecount
show that f is continuous at (-infinity, infinity)

f(x)= { sin(x) if x < (pi/4)
cos(x) if x >or equal to (pi/4)

confused???
$\displaystyle f\left( x \right) = \left\{ \begin{gathered} \sin x,{\text{ if }}x < \frac{\pi } {4} \hfill \\ \cos x,{\text{ if }}x \geqslant \frac{\pi } {4} \hfill \\ \end{gathered} \right. \hfill \\$

$\displaystyle f\left( x \right){\text{ is continuous in }}\left( { - \infty ,{\text{ }}\infty } \right){\text{ if it is continuous at }}x = \frac{\pi } {4}. \hfill \\$

$\displaystyle {\text{For }}f\left( x \right){\text{ to be continuous at }}x = \frac{\pi } {4}, \hfill \\$

$\displaystyle \mathop {\lim }\limits_{x \to \frac{\pi } {4} - } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi } {4} + } f\left( x \right) = f\left( {\frac{\pi } {4}} \right) \hfill \\$
$\displaystyle {\text{Now, Left hand limit, }} = \mathop {\lim }\limits_{x \to \frac{\pi } {4} - } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi } {4} - } \sin x = \sin \frac{\pi } {4} = \frac{1} {{\sqrt 2 }} \hfill \\$

$\displaystyle {\text{Right hand limit, }} = \mathop {\lim }\limits_{x \to \frac{\pi } {4} + } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi } {4} + } \cos x = \cos \frac{\pi } {4} = \frac{1} {{\sqrt 2 }} \hfill \\$

$\displaystyle f\left( {\frac{\pi } {4}} \right) = \cos \frac{\pi } {4} = \frac{1} {{\sqrt 2 }} \hfill \\$

$\displaystyle {\text{Since, }}\mathop {\lim }\limits_{x \to \frac{\pi } {4} - } f\left( x \right) = \mathop {\lim }\limits_{x \to \frac{\pi } {4} + } f\left( x \right) = f\left( {\frac{\pi } {4}} \right) = \frac{1} {{\sqrt 2 }}\hfill \\$

$\displaystyle {\text{So, the function is continuous at }}x = \frac{\pi } {4} \hfill \\$

$\displaystyle {\text{So, the function }}f\left( x \right){\text{ is continuous in }}\left( { - \infty ,{\text{ }}\infty } \right) \hfill \\$