Aspire's Library
A Place for Latest Exam wise Questions, Videos, Previous Year Papers,
Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let $ f(x) = \lim_{\theta \to 0} \left( \frac{\cos \pi x - x^{(\theta)} \sin(x-1)}{1 + x^{(\theta)} (x - 1)} \right),; x \in \mathbb{R}. $
Consider the following two statements :
(I) $ f(x) $ is discontinuous at $ x = 1 $.
(II) $ f(x) $ is continuous at $ x = -1 $.
Then,
Solution
$ f(x) = \begin{cases} \cos \pi x, & x \to 1^- \ -\frac{\sin(x-1)}{(x-1)}, & x \to 1^+ \end{cases} $
RHL $ = \lim_{x \to 1} -\frac{\sin(x-1)}{(x-1)} = -1 $
LHL $ = \lim_{x \to 1} \cos \pi x = -1,; f(1) = -1 $
$ f(x) $ is continuous at $ x = 1 $
$ f(x) = \begin{cases} -\frac{\sin(x-1)}{(x-1)}, & x \to -1^- \ \cos \pi x, & x \to -1^+ \end{cases} $
RHL $ = \lim_{x \to -1} \cos \pi x = -1 $
LHL $ = \lim_{x \to -1} -\frac{\sin(x-1)}{(x-1)} = \frac{\sin 2}{2} $
$ f(x) $ is discontinuous at $ x = -1 $
Online Test Series, Information About Examination,
Syllabus, Notification
and More.
View More
Online Test Series, Information About Examination,
Syllabus, Notification
and More.
View More
Ask Your Question or Put Your Review.
