Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
0 votes

Discuss the continuity of the following functions $(b)\;f(x)=\sin x-\cos x$

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com
Can you answer this question?

1 Answer

0 votes
  • $\sin(a-b)=\sin a\cos b-\cos a\sin b$
  • If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
$f(x)=\sin x-\cos x$
Multiply and divide by $\sqrt 2$
$\qquad=\sqrt 2[\large\frac{1}{\sqrt 2}$$\sin x-\large\frac{1}{\sqrt 2}$$\cos x]$
$\qquad=\sqrt 2[\sin x\cos\large\frac{\pi}{4}$$-\cos x\sin\large\frac{\pi}{4}]$
The above equation is of the form $\sin(a-b)=\sin a\cos b-\cos a\sin b$
$\Rightarrow \sqrt 2\sin(x-\large\frac{\pi}{4})$
Step 2:
At $x=a$ where $a\in R$
LHL=$\lim\limits_{\large x\to a^-}\sqrt 2\sin(x-\large\frac{\pi}{4})$
$\qquad=\lim\limits_{h\to 0}\sqrt 2\sin(a-h-\large\frac{\pi}{4})$
$\qquad=\lim\limits_{h\to 0}\sqrt 2[\sin(a-\large\frac{\pi}{4})$$\cos h-\cos(a-\large\frac{\pi}{4})$$\sin h$
$\sin(A-B)=\sin A\cos B-\cos A\sin B$
$\qquad=\sqrt 2[\sin(a-\large\frac{\pi}{4})$$\cos 0-\sqrt 2\cos(a-\large\frac{\pi}{4})$$\sin 0$
$\qquad=\sqrt 2\sin(a-\large\frac{\pi}{4})$
Step 3:
RHL=$\lim\limits_{\large x\to a^+}\sqrt 2\sin(x-\large\frac{\pi}{4})$
$\qquad=\lim\limits_{h\to 0}\sqrt 2\sin(a+h-\large\frac{\pi}{4})$
$\qquad=\lim\limits_{h\to 0}\sqrt 2[\sin(a-\large\frac{\pi}{4})$$\cos h+\cos(a-\large\frac{\pi}{4})$$\sin h$
$\qquad=\sqrt 2[\sin(a-\large\frac{\pi}{4})$$\cos 0-\cos(a-\large\frac{\pi}{4})$$\sin 0]$
$\qquad=\sqrt 2\sin(a-\large\frac{\pi}{4})$
Step 4:
Also $f(a)=\sqrt 2\sin(a-\large\frac{\pi}{4})$
$\therefore$ LHL =RHL=f(a).
Hence $f(x)$ is continuous at all points.
answered Sep 11, 2013 by sreemathi.v

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App